ACFT EQUIP PROB -- ACFT DAMAGED -- ACR HAD AN ENG OVERSPD AND BOTH ENGS OVERTEMPED DURING TKOF AND INITIAL CLB.

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WE DEPARTED COS ON A FLT BTWN COS-PHX ON AUG/TUE/95. DEP RWY WAS 17L WITH AN OUTSIDE AIR TEMP OF 65 DEGS F. WE WERE CLRED FOR TKOF; WE WERE ISSUED A 160 DEG MAGNETIC HDG FOR DEP TO HOT AIR BALLOON ACTIVITY S OF THE ARPT. AS WE STARTED OUR TKOF ROLL; THE AUTOTHROTTLES ADVANCED PWR TO MAX FOR THE TEMP (65 DEG F) SET IN THE FMC. AT APPROX 80 KTS; I NOTICED AN OVERSHOT ON #2 N1 OF APPROX 100.7 PERCENT; WITH THE TARGET BEING 97.3 PERCENT N1. I ATTEMPTED TO CORRECT THE PWR SETTING MANUALLY DURING TKOF. AS WE ROTATED WE NOTICED THE 2 BALLOONS HAD DRIFTED FURTHER E AND ADJUSTED OUR HDG TO CLR THEM. AFTER WE WERE CLR I LOOKED DOWN TO THE ENG INSTS AND NOTICED BOTH RED OVERTEMP LIGHTS WERE ON IN THE EGT GAUGES. AS IT WAS BUMPY; I NOTICED 965 DEGS ON #1 ENG AND APPROX 965-980 DEGS ON #2 ENG. WE IMMEDIATELY SELECTED CLB THRUST WHICH BROUGHT ALL READINGS BACK TO NORMAL. WE CONTINUED TO MONITOR THE ENG INSTS LOOKING FOR ANYTHING ABNORMAL WHICH WOULD HAVE DICTATED A RETURN TO COS; BUT EVERYTHING WAS NORMAL FOR THE REST OF THE FLT. WE ACCOMPLISHED THE APPROPRIATE CHKLISTS AND MADE LOGBOOK ENTRIES AS REQUIRED. INBOUND TO PHX; MAINT CTL WAS CONTACTED AND MADE AWARE OF WHAT TRANSPIRED. I THEN BRIEFED MAINT ON WHAT HAPPENED WHEN WE WERE AT THE GATE AND APPARENTLY BOTH ENGS HAD TO BE CHANGED. THERE WERE SEVERAL FACTORS WHICH CONTRIBUTED TO THE EGT OVERTEMP -- KEEPING VISUAL CONTACT WITH THE BALLOONS S OF THE ARPT. OUR EYES WERE OUTSIDE THE COCKPIT; OUR FIRST PRIORITY TFC AVOIDANCE. HAD WE BEEN ON INSTS; WE WOULD HAVE BEEN MONITORING ALL INSTS AND WOULD HAVE BEEN MONITORING ALL INSTS AND WOULD HAVE NOTICED THE HIGH EGTS. WE APPARENTLY ENCOUNTERED A TEMP INVERSION APPROX 1000-1500 FT AGL. AS THE AUTOTHROTTLES WERE IN THE HOLD MODE; IT PRECLUDED THE FMC FROM REDUCING PWR TO LOWER THE EGTS. THE APPROX TEMP INCREASE WAS PLUS 15-25 DEGS F OUTSIDE AIR TEMP WHICH WOULD CONTRIBUTE TO HIGHER EGTS. AGAIN; OUR EYES WERE OUTSIDE THE COCKPIT SO MOST ALL OF THE ABOVE WENT UNNOTICED UNTIL WE WERE WELL CLR OF ANY TFC.