The "smoothing" comes from a gap created by `Exclusions`. Sketch the pieces separately and combine:
 
 Show[
 Plot3D[Im@expr[z] /. z -> (x + I y), {x, -1/2, 3/2}, {y, -1/2, 
 0},(*Enhanced plot options*)PlotRange -> {-5, 5}, 
 PlotPoints -> 150,(*Increased for better resolution*)
 MaxRecursion -> 
 7,(*Higher recursion for better singularity capture*)(*Color scheme\
 that emphasizes extremes*)
 ColorFunction -> (ColorData["TemperatureMap"][
 Rescale[#3, {10, -10}, {0, 1}]] &), 
 ColorFunctionScaling -> False,(*Exclusions to handle singularities*)
 ExclusionsStyle -> {None, 
 Directive[Red, Thick]},(*Enhanced mesh for contour visualization*)
 MeshFunctions -> {#3 &}, Mesh -> {Range[-3, 3, .5]}, 
 MeshStyle -> {{Opacity[0.5], Darker[Blue]}, {Opacity[0.3], Gray}},
 (*Visual styling*)
 PlotStyle -> Directive[Specularity[White, 10], Opacity[0.95]], 
 ClippingStyle -> None, Exclusions -> {Im[Sqrt[x + I y]] == 0}, 
 Lighting -> {{"Ambient", GrayLevel[0.3]}, {"Directional", White, 
 ImageScaled[{0, 0, 2}]}, {"Directional", White, 
 ImageScaled[{1, 1, 1}]}},(*Labels and formatting*)
 PlotLabel -> 
 Style["Stieltjes Transform of Uniform", FontSize -> 16, 
 FontWeight -> Bold], 
 AxesLabel -> {Style["Re[z]", FontSize -> 14, Bold], 
 Style["Im[z]", FontSize -> 14, Bold], 
 Style["Im[f(z)]", FontSize -> 14, Bold]},(*View and layout*)
 BoxRatios -> {1, 1, 0.6}, ViewPoint -> {-2.5, 2.5, 1.8}, 
 ViewVertical -> {0, 0, 1}, 
 PlotTheme -> "Detailed",(*Additional styling*)
 Background -> GrayLevel[0.95], ImageSize -> Large, 
 AxesStyle -> Directive[Black, Thickness[0.002]], Boxed -> False, 
 FaceGrids -> None, BaseStyle -> {FontFamily -> "Helvetica"}],
 Plot3D[Im@expr[z] /. z -> (x + I y), {x, -1/2, 3/2}, {y, 0, 
 1/2},(*Enhanced plot options*)PlotRange -> {-5, 5}, 
 PlotPoints -> 150,(*Increased for better resolution*)
 MaxRecursion -> 
 7,(*Higher recursion for better singularity capture*)(*Color scheme\
 that emphasizes extremes*)
 ColorFunction -> (ColorData["TemperatureMap"][
 Rescale[#3, {10, -10}, {0, 1}]] &), 
 ColorFunctionScaling -> False,(*Exclusions to handle singularities*)
 ExclusionsStyle -> {None, 
 Directive[Red, Thick]},(*Enhanced mesh for contour visualization*)
 MeshFunctions -> {#3 &}, Mesh -> {Range[-3, 3, .5]}, 
 MeshStyle -> {{Opacity[0.5], Darker[Blue]}, {Opacity[0.3], Gray}},
 (*Visual styling*)
 PlotStyle -> Directive[Specularity[White, 10], Opacity[0.95]], 
 ClippingStyle -> None, Exclusions -> {Im[Sqrt[x + I y]] == 0}, 
 Lighting -> {{"Ambient", GrayLevel[0.3]}, {"Directional", White, 
 ImageScaled[{0, 0, 2}]}, {"Directional", White, 
 ImageScaled[{1, 1, 1}]}},(*Labels and formatting*)
 PlotLabel -> 
 Style["Stieltjes Transform of Uniform", FontSize -> 16, 
 FontWeight -> Bold], 
 AxesLabel -> {Style["Re[z]", FontSize -> 14, Bold], 
 Style["Im[z]", FontSize -> 14, Bold], 
 Style["Im[f(z)]", FontSize -> 14, Bold]},(*View and layout*)
 BoxRatios -> {1, 1, 0.6}, ViewPoint -> {-2.5, 2.5, 1.8}, 
 ViewVertical -> {0, 0, 1}, 
 PlotTheme -> "Detailed",(*Additional styling*)
 Background -> GrayLevel[0.95], ImageSize -> Large, 
 AxesStyle -> Directive[Black, Thickness[0.002]], Boxed -> False, 
 FaceGrids -> None, BaseStyle -> {FontFamily -> "Helvetica"}], 
 ParametricPlot3D[{{x, 0, 
 Im@expr[z] /. z -> (x + I (-$MachineEpsilon))}, {x, 0, 
 Im@expr[z] /. z -> (x + I (+$MachineEpsilon))}}, {x, -1/2, 3/2}, 
 PlotStyle -> Directive[Red, Thick]],
 PlotRange -> All]

[![enter image description here][1]][1]

---

@azerbajdzan's modification of the singular locus can be implemented in the method in my comment. Since the method is a simpler approximation, I thought I'd include both:

 expr[z_] = 
 Expectation[1/(z - x), x \[Distributed] UniformDistribution[]];
 ParametricPlot3D[
 {x, y^3/(1/2)^2, Im@expr[z] /. z -> (x + I y^3/(1/2)^2)},
 {x, -1/2, 3/2}, {y, -1/2, 1/2},(*Enhanced plot options*)
 PlotRange -> {Automatic, Automatic, {-5, 5}}
 , PlotPoints -> 25
 , MaxRecursion -> 4
 ,(*Color scheme that emphasizes extremes*)
 ColorFunction -> (ColorData["TemperatureMap"][
 Rescale[#3, {4, -4}, {0, 1}]] &), 
 ColorFunctionScaling -> False,(*Exclusions to handle singularities*)
 ExclusionsStyle -> {None, 
 Directive[Red, Thick]},(*Enhanced mesh for contour visualization*)
 MeshFunctions -> {#3 &}, Mesh -> {Range[-3, 3, .5]}, 
 MeshStyle -> {{Opacity[0.5], Darker[Blue]}, {Opacity[0.3], 
 Gray}},(*Visual styling*)
 PlotStyle -> Directive[Specularity[White, 10], Opacity[0.95]]
 (*,ClippingStyle->None*)
 , Exclusions -> {{Im[z] == 0, 0 <= Re[z] <= 1} /. z -> x + I y}
 , Lighting -> {{"Ambient", GrayLevel[0.3]}, {"Directional", White, 
 ImageScaled[{0, 0, 2}]}, {"Directional", White, 
 ImageScaled[{1, 1, 1}]}},(*Labels and formatting*)
 PlotLabel -> 
 Style["Stieltjes Transform of Uniform", FontSize -> 16, 
 FontWeight -> Bold], 
 AxesLabel -> {Style["Re[z]", FontSize -> 14, Bold], 
 Style["Im[z]", FontSize -> 14, Bold], 
 Style["Im[f(z)]", FontSize -> 14, Bold]},(*View and layout*)
 BoxRatios -> {1, 1, 0.6}, ViewPoint -> {-2.5, 2.5, 1.8}, 
 ViewVertical -> {0, 0, 1}, 
 PlotTheme -> "Detailed",(*Additional styling*)
 Background -> GrayLevel[0.95], ImageSize -> Large, 
 AxesStyle -> Directive[Black, Thickness[0.002]], Boxed -> False, 
 FaceGrids -> None, BaseStyle -> {FontFamily -> "Helvetica"}]

[![enter image description here][2]][2]


 [1]: https://i.sstatic.net/LRHORb0d.png
 [2]: https://i.sstatic.net/iMer4Fj8.png