In a large skillet with a lid, heat the olive oil and 1 tablespoon butter over medium heat. Add 1 tablespoon thyme, and 1/2 teaspoon of salt and pepper. Add the potatoes and toss gently to coat. Arrange the potatoes cut side down, then cover and cook for 10 minutes, stirring halfway through. Remove potatoes to a clean plate, tent loosely and set aside.
For the chicken:
While the potatoes are cooking, prepare the chicken. Dry chicken breasts with paper towels. Season generously with salt and pepper.
Heat 2 tablespoons butter in the now-empty skillet. Add the chicken and cook until well browned on one side, 6 to 8 minutes. Turn the chicken and sprinkle with the garlic and 2 teaspoons fresh thyme leaves. Cover each breast with sliced lemon.
Push the chicken to one side the skillet. On the empty side of the skillet, add the remaining 1 tablespoon butter, garlic and crushed red pepper flakes. Add the potatoes back into the pan and toss in the garlic butter sauce. Cover and cook for 10 minutes, or until the chicken is cooked through (160°F) and the potatoes are tender.
Drizzle the chicken and potatoes with fresh lemon juice and garnish with fresh thyme sprigs. Serve immediately.
Substitute boneless, skinless chicken thighs if desired.