Most (residential) neighborhood Chinese places will let you do this even if it isn’t explicitly listed on the menu (as in, may I please have chicken and broccoli but with black bean sauce instead of brown or white, and please add mushroooms and snow peas too).
But if you are looking for explicit menu listings, here are two in the same neighborhood.
Many Sichuan places have the option of dry pot where you choose your own ingredients, but the DIY option from Legend seems set up to cater to the non-Sichuan seeking neighborhood audience that other places are capturing – hence the standard Cantonese white, brown, garlic, and black bean sauce options.
