A B737 CREW DISTRACTED BY A HIGH ENG EGT; ALLOWED THE HDG TO DRIFT AFTER TKOF RESULTING IN ATC STOPPING THEIR CLB AND GIVING VECTORS.

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THE ACFT HAD A HISTORY OF HIGH EGT ON THE #1 ENG DURING TKOF AND CLBOUT. IT HAD BEEN WRITTEN UP IN THE LOGBOOK AND THE FO FROM THE PREVIOUS CREW BRIEFED ME ABOUT IT ON THE TURNOVER. HE HAD ALSO SAID HIS CAPT HAD TALKED TO MAINT ABOUT IT. THIS WAS THE THIRD LEG WITH THIS ACFT. ON THE PREVIOUS 2 LEGS THE EGT HAD INDEED CLBED UNUSUALLY HIGH BUT WITHIN LIMIT READINGS. ON THIS TKOF/CLBOUT THE EGT HAD REACHED ITS HIGHEST READING OF JUST OVER 925 DEGS AND JUST UNDER THE RED LINE; HIGHER THAN ON THE 2 PREVIOUS FLTS. I RECALL IT REACHING APPROX 880 DEGS AT VR AND CONTINUED TO CLB AFTER ROTATION. IT CONTINUED AT A RATE THAT GAVE ME AN EXPECTATION THAT IT WOULD CLB THROUGH REDLINE. I REDUCED THE THRUST LEVERS BELOW CLB THRUST WHICH REDUCED AND FINALLY STABILIZED THE EGT AT JUST OVER 925 DEGS. THE CONCERN WITH THE EGT ALLOWED MY SCAN TO BE DIVERTED FROM MY DEP HDG ON THE DEP. PASSING APPROX 3000 FT MSL; DEP TOLD US TO LEVEL OFF AT 4000 FT MSL AFTER WE HAD BEEN ASSIGNED 10000 FT MSL. THEY HAD INQUIRED AS TO WHY WE WERE HDG NE VERSUS N AND THAT THERE WAS ANOTHER ACFT IN THE PATTERN AT 5000 FT MSL. I RETURNED TO THE 360 DEG HDG. THEY REASSIGNED 10000 FT MSL AND WE RESUMED THE DEP. THEY DID NOT GIVE US A NUMBER TO CALL AND WE HEARD NOTHING FURTHER FROM ATC; NOR DID WE RECEIVE A TCAS ALERT. AVIATE; NAV; COM. THE CONCERN WITH THE POTENTIAL OVERTEMP WAS CERTAINLY LEGITIMATE BUT THERE WAS NO EXCUSE FOR IT TAKING PRECEDENT OVER FLYING THE ACFT. THOUGH BOTH MY FO AND I WERE AWARE OF THE EGT ISSUE; I BELIEVE IT DREW BOTH OUR ATTN MOMENTARILY AND ENABLED A DRIFT OF THE ACFT GOING UNNOTICED FOR 3-5 SECONDS. CONSEQUENTLY; I SHOULD HAVE MORE THOROUGHLY BRIEFED A DIVISION OF DUTIES AS TO XCHKING FLT INSTS AND ENG READINGS. SUPPLEMENTAL INFO FROM ACN 779113: THE FLT CONTINUED NORMALLY FROM THAT POINT ON. I AM NOT SURE WHERE THE BREAKDOWN OCCURRED; SO IT IS HARD TO PINPOINT ANY CHANGES THAT SHOULD BE MADE. MAYBE A MORE IN-DEPTH REVIEW OF THE PDC BY BOTH PLTS BEFORE TKOF.