A center FCC failure during a high performance climb to an initial level off altitude contributed to a momentary loss of control for a B757-200 flight crew.

Narrative

Due to use of engine anti-ice and a heavy aircraft that precluded use of D2; we performed a max takeoff off 32R at ZZZZ. Takeoff roll and lift off were normal. At 1500 feet; I called for VNAV and then the center autopilot per SOP. I sensed the aircraft was climbing rapidly due to the max takeoff and called for CLB 2. I remember ALT CAP on the FMA [Flight Mode Annunciator] occurred fairly early; and at some point they indicated SPD and ALT CAP. At some point it went to ALPHA and a line through ALT CAP. At approximately 2000 FT; the aircraft abruptly continued to nearly 30 degrees nose high with airspeed decaying rapidly. I did not catch this immediately as I had turned my focus to the sensitive departure. As I returned my scan to the ADI; I saw almost 30 degrees nose high; and disconnected the autopilot; lowered the nose; and fire walled the thrust. At some point during the event we lost both flight directors. Stick shaker activation occurred twice. At the peak of the nose up attitude; and again when I lowered the nose to 10 degrees. Once the aircraft was recovered and stable; we reselected the center autopilot and the flight continued uneventfully. I am very comfortable saying that I am absolutely sure we used the correct procedures for the departure. I am 100% sure I called for VNAV and then the autopilot in that sequence; and I know my First Officer did exactly as I commanded. I feel I was slow to get the autopilot off due to glancing to the departure page after I thought the FMAs were correct. I was not prepared for the aircraft to respond the way it did; and when I came back to the instruments; it was rapidly heading toward an unusual attitude and I was not sure whether I had a pitot-static issue due to deicing and the aircraft sitting over the weekend; some kind of ADI problem; or some other issue. I finally determined the standby ADI was correct; and flew the recovery on that instrument. I also probably should have used a different autopilot after the aircraft had stabilized; but it took us a long time just to determine what put us into that situation due to the sheer rapidity of the event. That morning; after I rested; I called System Operations back; and they informed me that the center FCC had failed a check and been deferred. We flew the aircraft back to CGN uneventfully the next evening.