Season the breasts with salt and pepper on both sides.
Preheat oven to 350 degrees.
Heat the oil in a large skillet over media-high to high heat. Place the breasts in and sear for a couple of minutes on both sids until golden. Pop the pan into the oven for 10-15 minutes more until the chicken is cooked through. Remove the chicken from the pan and place on a plate, tented with foil.
Heat the pan on the stove over medium-high heat. Add in the shallot and garlic and stir until fragrant, about 30 seconds. Add in the shiitakes and cook until tender, about 3 minutes. Pour in the vermouth, and cook, scraping up all of the browned bits on the bottom on the pan for about a minute. Pour in the broth and simmer until reduced by half. Stir in the cream, scallions, and thyme. Add the chicken back into the pan and coat with the sauce. Serve.
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