letting-in two-ply 2x12 girder to 6x6 beam
I've done that before many times on large beams and it works fine. In fact, forgo the reciprocating saw and still use your chisel. It'll easier than the other method because you'll be working with the grain instead of against it. Or you could use the reciprocating saw to finished the depth on   your cross cuts for your usual method.

The brace is called a knee brace and it's often of evenly dimensioned lumber, usually 4x4, and is usually attached on the face of the beam and post rather than the side. Lag screws are used for attachment, through the knee brace at 45 degree angle which is at the normal 90 degree angle into the post and beam. Usually a counter bore is used to both hide the screw head from view from the side and to provide a flat bearing surface