Telehealth Extension and Evaluation Act This bill expands and otherwise modifies coverage of telehealth services under Medicare until two years after the end of the COVID-19 public health emergency. Specifically, the bill (1) allows federally qualified health centers and rural health clinics to serve as the distant site (i.e., the location of the health care practitioner), (2) allows for Medicare payment of certain audio-only services and of outpatient critical access hospital services consisting of telehealth behavioral therapy, (3) conditions payment for certain high-cost laboratory tests and durable medical equipment that are ordered via telehealth on at least one in-person visit during the preceding 12-month period, and (4) allows Schedule II through V controlled substances to be prescribed online if a practitioner has conducted a telehealth evaluation with video. The bill also generally extends any Medicare telehealth flexibilities that were granted during the COVID-19 public health emergency until two years after the emergency ends. The Centers for Medicare & Medicaid Services (CMS) must report on the effects of changes that were made during the emergency period with respect to the provision or availability of telehealth services under Medicare. The CMS must also award grants to state Medicaid programs to allow them to report on similar information.