With vanilla Emacs there's no way that I'm aware of, as @db38x@db48x reported.
However, here are 3 libraries that let you do it: La Carte, Key See, and Icicles.
La Carte
Use completion to match anything in any menu-bar menu, anywhere, anytime. Doesn't matter where you are currently - what the current context is. You always have the entire menu-bar forest of menus available to you, and you can always match anywhere - any submenu or any menu item - in any of those menus.
Key See
Use completion to match any key bindings, including menu-bar bindings. Command kc-complete-menu-bar limits candidates to the latter (i.e., not keyboard key bindings).
Completion candidates at any time are at a single menu level. But you can go back up a level by choosing the special candidate ...
Icicles
This gives you everything that Key See gives you, and more.
And it enhances the behavior of La Carte, giving you much more control over exploring the menu forest. Key features in this regard are progressive completion, sorting of candidates, more powerful matching, showing candidate help, and filtering history.