Final answer:
The most immediate action is required for a new-onset change in the patient's voice, which could signal a serious complication like compression of the laryngeal nerve.
Step-by-step explanation:
When assessing a patient with Hashimoto's thyroiditis and a goiter, the finding that would require the most immediate action is 'a new-onset change in the patient's voice.' This could indicate a complication such as compression of the laryngeal nerve leading to vocal cord paralysis, which might require urgent evaluation and intervention. An apical pulse rate at rest of 112 beats/minute suggests tachycardia and may be concerning, but not as immediately urgent as potential airway compromise. An elevation in the patient's T3 and T4 levels would represent a hyperthyroid state, which might need treatment but again is not immediately life-threatening. A bruit audible bilaterally over the thyroid gland indicates increased blood flow, which can be found in hyperthyroidism or large goiters, but does not require immediate action. Therefore, the most urgent finding is the new-onset change in the patient's voice.