Can you do special blood test just for a the thyroid?

Yes. A single test can be done on a blood sample to start the investigation. Pcp can do it.
Yes. Your primary care doctor can do that.
Yes. A thyroid panel consisting of tsh, t4, and free T3 (liothyronine) can give a lot of information regarding thyroid function.
Yes. The specialist to see for potential thyroid issues is an endocrinologist.