This clinical trial studies how well thyroid lobectomy with ipsilateral central neck dissection works to treat medullary thyroid cancer (MTC) in patients without a germline RET mutation (sporadic). Currently, sporadic and germline RET (rearranged during transfection) mutation positive MTCs that are limited to the thyroid are managed in the same way, complete surgical removal of the entire thyroid gland (total thyroidectomy) with surgical removal of lymph nodes and other tissues on both sides of the neck (bilateral central neck dissection). Total thyroidectomy and bilateral central neck dissection carry a high risk of complications, and total thyroidectomy requires patients to take lifelong thyroid hormone replacement therapy, which can impact quality of life. Research has shown that patients with sporadic MTC do not have a high risk of developing MTC in the remaining normal thyroid tissue and that they may be able to be managed differently than patients with germline RET mutations. Thyroid lobectomy with ipsilateral central neck dissection is a surgical procedure which removes only the lobe of the thyroid gland that is affected by cancer as well as the lymph nodes and other tissues from the affected side of the neck. Thyroid lobectomy with ipsilateral central neck dissection may be a safer, more tolerable, and/or more effective way to treat sporadic MTC.