On our gas oven at home we had the exact same issue. It was a relay that senses the oven temp. Ours would do exactly what you described. When first starting it would come up to the temp called for, but the burner would not come back on without turning the oven completely off.
Went to the local RV parts place, bought a new sensor and replaced it. The sensor/relay on our was under the oven where the burner sits. Once I had the part it took about 30-40 minutes to replace and get the oven working again.
The part itself wasn't expensive, maybe about $15CDN.
