Mnemonic anchor: Hashimoto equals Hurthle cells plus High TSH.
Read the labs first: TSH is elevated, so the gland is underactive. That single fact removes Graves disease, where TSH is suppressed because the gland is overdriven.
Read the histology: dense lymphocytic infiltrate with germinal centres and Hurthle (oxyphil) cells is the signature of chronic autoimmune thyroiditis. Cancers behave differently: follicular carcinoma is diagnosed by capsular or vascular invasion, and medullary carcinoma shows neuroendocrine C cells with amyloid stroma, neither of which is described.
Putting it together: a hypothyroid young woman with lymphocytic, Hurthle-cell-rich thyroid tissue has Hashimoto's thyroiditis, the leading cause of acquired hypothyroidism. Anti-TPO antibodies would confirm it clinically.
Reference: standard endocrine pathology.