JavaScript (ES6), 141 138 bytes

String and direction are input via currying syntax.
'/\n|-DI>-/---< ' for left, '/\n|-DI<-\\---> ' for right.

t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}/ |-D${p}${[...t].join` `} ${p[7]} ${s}\\${_}\\`,p>'I>'?r:[...r].reverse().join``) 

f= t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}${p}${[...t].join` `} ${d=p[12]} ${s}\\${_}\\`,d>'<'?[...r].reverse().join``:r) console.log(f('HAPPY BIRTHDAY!')('/\n|-DI>-/---< ')) console.log(f('HAPPY BIRTHDAY!')('/\n|-DI<-\\---> '))

JavaScript (ES6), 141 138 bytes

String and direction are input via currying syntax.
'/\n|-DI>-/---< ' for left, '/\n|-DI<-\\---> ' for right.

t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}${p}${[...t].join` `} ${d=p[12]} ${s}\\${_}\\`,d>'<'?[...r].reverse().join``:r) 

f= t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}${p}${[...t].join` `} ${d=p[12]} ${s}\\${_}\\`,d>'<'?[...r].reverse().join``:r) console.log(f('HAPPY BIRTHDAY!')('/\n|-DI>-/---< ')) console.log(f('HAPPY BIRTHDAY!')('/\n|-DI<-\\---> '))

JavaScript (ES6), 141 138 bytes

String and direction are input via currying syntax.
'/\n|-DI>-/---< ' for left, '/\n|-DI<-\\---> ' for right.

t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}/ |-D${p}${[...t].join` `} ${p[7]} ${s}\\${_}\\`,p>'I>'?r:[...r].reverse().join``) 

f= t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}${p}${[...t].join` `} ${d=p[12]} ${s}\\${_}\\`,d>'<'?[...r].reverse().join``:r) console.log(f('HAPPY BIRTHDAY!')('/\n|-DI>-/---< ')) console.log(f('HAPPY BIRTHDAY!')('/\n|-DI<-\\---> '))

JavaScript, 4 bytes

Probably a loophole. Input a program that generates a left-facing plane for left, and a program that generates a right-facing plane for right.

eval 

JavaScript (ES6), 141 138 bytes

String and direction are input via currying syntax.
'/\n|-DI>-/---< ' for left, '/\n|-DI<-\\---> ' for right.

t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}/ |-D${p}${[...t].join` `} ${p[7]} ${s}\\${_}\\`,p>'I>'?r:[...r].reverse().join``) 

f= t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}${p}${[...t].join` `} ${d=p[12]} ${s}\\${_}\\`,d>'<'?[...r].reverse().join``:r) console.log(f('HAPPY BIRTHDAY!')('/\n|-DI>-/---< ')) console.log(f('HAPPY BIRTHDAY!')('/\n|-DI<-\\---> '))

JavaScript (ES6), 141 bytes

String and direction are input via currying syntax.  'I>-/---< ' for left, 'I<-\\---> ' for right.

t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}/ |-D${p}${[...t].join` `} ${p[7]} ${s}\\${_}\\`,p>'I>'?r:[...r].reverse().join``) 

f= t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}/ |-D${p}${[...t].join` `} ${p[7]} ${s}\\${_}\\`,p>'I>'?r:[...r].reverse().join``) console.log(f('HAPPY BIRTHDAY!')('I>-/---< ')) console.log(f('HAPPY BIRTHDAY!')('I<-\\---> '))

JavaScript (ES6), 141 138 bytes

String and direction are input via currying syntax. 
'/\n|-DI>-/---< ' for left, '/\n|-DI<-\\---> ' for right.

t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}/ |-D${p}${[...t].join` `} ${p[7]} ${s}\\${_}\\`,p>'I>'?r:[...r].reverse().join``) 

f= t=>p=>(r=`${s=' '}/${_='-'.repeat(t.length*2+1)}${p}${[...t].join` `} ${d=p[12]} ${s}\\${_}\\`,d>'<'?[...r].reverse().join``:r) console.log(f('HAPPY BIRTHDAY!')('/\n|-DI>-/---< ')) console.log(f('HAPPY BIRTHDAY!')('/\n|-DI<-\\---> '))

JavaScript, 4 bytes

Probably a loophole. Input a program that generates a left-facing plane for left, and a program that generates a right-facing plane for right.

eval