。
She earned a reputation for being thoroughly versed in her field.
Literal
She [topic-は] her own specialty [matter-こと] [regarding-に関して] expert-knowledgeable-is [quotative-という] evaluation [object-を] received.
~に関して ('regarding, concerning, with respect to') is more formal than ~について; ~のこと softens 専門 ('specialty') into 'matter of one's field.' 精通している ('to be thoroughly versed in, to have detailed knowledge of') is a Sino-Japanese verb belonging to written/formal registers. ~という評価を受ける ('to receive an evaluation that X / earn a reputation for X') uses という to wrap the entire characterization as a quoted label attached to 評価 ('evaluation, reputation').