In a large skillet, heat 1 tablespoon of the oil over medium heat. Sprinkle both sides of chicken with salt and fresh pepper. Add chicken to hot pan. Cook for 8 to 10 minutes or until chicken pieces are well browned, flipping once to brown both sides evenly. Do not try to flip chicken too early or it will stick in the pan.

Remove skillet from the heat. Carefully add chicken broth to the skillet. Return to the heat. Bring to boiling; reduce heat. Simmer, covered, for 8 to 12 minutes or until chicken is no longer pink (165°F when tested with an instant-read thermometer). Transfer chicken to a plate; cover to keep warm. Reserve ¼ cup of the cooking liquid (note: if you don’t have ¼ cup liquid left in the skillet, add enough additional chicken broth to make ¼ cup); set aside. Discard any remaining cooking liquid from the skillet.

In the same skillet, cook mushrooms in remaining 1 tablespoon oil over medium heat for 6 to 8 minutes or until mushrooms are just tender and golden brown, stirring occasionally. Stir in sliced garlic; cook about 30 seconds. Remove skillet from the heat. Carefully pour in wine. Return to medium heat. Cook, uncovered, 3 to 4 minutes or until most of the liquid is evaporated.

In a small bowl whisk together coconut milk, mustard, and arrowroot until completely smooth. Add all at once to skillet with mushrooms. Add reserved ¼ cup chicken cooking liquid. Cook and stir mushrooms and sauce over medium heat until just boiling; reduce heat. Cook and stir 1 minute more. Season sauce to taste with coarse salt and fresh pepper.

Add cooked chicken to sauce in skillet. Sprinkle with thyme. Turn chicken pieces to coat with sauce. Cook about 1 minute more to make sure chicken is hot. Serve chicken topped with mushrooms and sauce.