How would you go about cutting a notch in a regular 2x4 stud
across the grain, so that the notch is 1.5" into the grain
and 3.5" along it? (i.e. to embed one 2x4 crosswise into
another).

1. Hold the board you want to inset against the studs and trace the sides
of the board on the stud(s).

2. Lay the board on the floor, set the circular saw on top, and adjust the
depth of cut so it matches the thickness of the board.

3. Cut the stud(s) at the upper and lower mark, and then make a few passes
in between. I usually space the cuts about 1/4" to 1/2" apart. It takes a
few more seconds, but it's easier to knock the pieces out and produces a
bit cleaner notch.

4. Take a hammer and knock out the pieces. You can clean up the notch with
the claw of the framing hammer, or use a chisel if you really want to be
neat.

We built our own house last year and had to inset a lot of diagonal braces
in the walls. So I notched a LOT of studs like I described above.

Depending on your intended function, you might just install blocking
"Between" the studs. I had to install a lot of blocking in our master
bedroom to nail up T&G pine boards vertically. In that situation it was
easier to put the blocking in and attach it with screws, especially since
the insulation and wiring was already in place.

Anthony