Solid oxide electrolysis cells (SOECs) offer high-efficiency hydrogen production but face significant thermal operability challenges under varying loads. This study develops a system-level optimization framework to identify feasible high-efficiency SOEC operating maps considering constraints on axial thermal gradients, maximum cell temperature, and inlet gas-cell temperature difference. A physics-based SOEC model is integrated with balance-of-plant models and an adaptive Gaussian-process-based optimization algorithm. Results show that the baseline system achieves a maximum efficiency of 88.3% at 40 A, close to thermoneutral operation. The dominant thermal constraint shifts from inlet temperature difference at low loads to axial thermal gradient at high loads. Increasing operating temperature shifts the optimal load upward, whereas degradation shifts it downward and narrows the feasible operating range. Under a real fluctuating wind power profile, enforcing thermal constraints reduces total hydrogen production by 5.3% in the baseline case and by 19.8% and 24.4% under high degradation and low-temperature operation respectively. This highlights the importance of considering thermal operability when optimizing the dispatch of an SOEC system for hydrogen production.