We did both last week (CM for Thanksgiving dinner, and CP for breakfast on Friday) and had a great time. Both are buffets, of course, unlike the plated breakfast at CRT, so you have more choices and can eat as much or as little as you want. Chef Mickey's seems much more hectic, which can be fun if you're in the mood, but I also can see how it would be overwhelming. CP was bustling, but I didn't get the same sense there.
We had a 9:15 a.m. breakfast at CP, so we didn't get out into the park until after 10:00 a.m. Even so, we walked over to Frontierland and found only a 10 minute wait at Splash Mountain. Adventureland wasn't bad either; no waiting afor the Tiki Room. Although things did get more crowded Pirates experienced technical difficulties and was shut down. Fantasyland was more of a problem, although we walked right into Mickey's Philharmagic. Tomorrowland was crowded and we had a long standby wait at SM, but we got BL Fast Passes when we got in line at SM and went right over there after a brief wait for our return time.
I might suggest avoiding an in-park character breakfast on an early entry day, however.