Hi Kong,
Plunging and machining a blind recess in the lathe is challenging, especially the roughing. I would attempt to do the basic roughing in the mill and then take a light cut in the lathe with a profiling tool. You can buy boring bars that are made for profiling. I would check out Kennametal (as an example) for ideas: Top Notch Profiling Boring bar A08-NKQCR05 This is a 1/2" shank bar which will bore in a hole as small as .67". But, if properly set on center, I would expect that you could face right to X0 with it, at about .005" depth of cut. This bar has 17 degrees of clearance between the flank of the insert and the part face.
First you get good, then you get fast. Then grouchiness sets in.
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)