Extrude along Surface

Hi, I’m trying to “engrave” the gray extrusion, but I can’t find a way to extrude the green grid that allows me to do so. I’ve tried by etruding the curves to point but in order to have a correct direction they will have to go through the object. Any suggestion?

I think something like this would do it.

Take a copy of your top grey surface. Split it with the squares, so you get the little squares left over. Use OffsetSrf (BothSides=Yes Solid=Yes) on the squares to create something that can punch through. Use BooleanDifference to the subtract those thickened squares. It’s kind of brute force, but might help.