Place each chicken breast between plastic wrap and gently pound or cut to about 1/2-inch thickness so they cook evenly.
Season both sides of the chicken with 1/2 teaspoon garlic powder, 1 teaspoon dried Italian seasoning, salt, and pepper to taste.
Heat 1 tablespoon olive oil in a large skillet over medium-high heat until shimmering.
Add the chicken to the skillet and cook about 6 minutes per side, or until cooked through and no longer pink; transfer to a plate and set aside.
Reduce heat to medium, add 2 tablespoons butter to the skillet, then sauté the minced shallot until softened, about 2–3 minutes.
Pour in 1 cup chicken broth and add the softened package of Boursin cheese; stir until the cheese melts and forms a creamy sauce.
Stir in 1/2 cup heavy cream and 1/2 cup grated Parmesan; whisk and simmer until the sauce is heated through and slightly thickened, about 4–5 minutes.
Return the cooked chicken to the skillet and warm in the sauce for 1–2 minutes to reheat and coat the chicken.
Adjust seasoning with salt and pepper if needed, then serve topped with chopped fresh parsley and thyme.