Season the chicken breasts with salt and pepper. Heat olive oil in a skillet over medium heat and add the chicken. Cook until browned on both sides and cooked through, about 6-7 minutes per side. Remove chicken and set aside.
Add minced garlic to the skillet and sauté for 1 minute. Pour in chicken broth and Worcestershire sauce, scraping the bottom of the skillet. Add thyme and butter, and stir until combined. Return the chicken to the skillet and let simmer.
Meanwhile, boil the quartered potatoes in a pot of salted water until tender, about 15 minutes. Drain and return to the pot. Add milk, cheddar cheese, and sour cream. Mash until creamy and smooth.
Serve the chicken with garlic gravy over the cheesy mashed potatoes. Garnish with fresh parsley.
Notes
You can adjust the thickness of the gravy by adding more or less chicken broth.