I traced your original bowl, combining the body of the bowl and the stem into a single profile. Notice, I could have made the bottom of the stem concave or recessed just by subtracting that from the profile.
I’m not sure what “it” is here. As for the logo/monogram, I copied the face from yours (cleaning it up in a few places). I extruded it only far enough to pass comfortably through one wall.
For one thing, as I mentioned, I didn’t plan to perform the same intersection four times (that’s the most error-prone operation), so I thought it made better sense to incline the thing so it passed more or less squarely through the wall of the cup instead of at an oblique angle as it did when you ran the same extrusion through both sides of the cup.
After intersecting the monogram with the quarter cup, I created a radial array of the quarter cup, restoring the other three quarters with the monogram pushed through.
Incidentally, when you refer to “engraving,” I generally take that to mean that some character or shape is recessed into a surface, not pushed all the way through. I guess I’m not sure whether that’s what you wanted to do. Although I could decidedly see all the way through yours, I guess I don’t know if that was deliberate. If you actually just want the monogram engraved, (pushed into but not through the surface) then obviously that little chunk won’t fall out.