Place peeled and cubed potatoes in a large pot of cold salted water. Bring to a boil, then reduce heat and simmer for 15-20 minutes until fork-tender.
Season chicken breasts on both sides with salt and black pepper.
Heat olive oil in a large skillet over medium-high heat. Sear chicken for 6-8 minutes per side until the internal temperature reaches 165°F (74°C). Remove chicken from the pan and set aside.
In the same skillet, melt 1 tablespoon of butter. Add sliced mushrooms and sauté for 5 minutes until browned. Stir in the minced garlic and cook for 1 minute until fragrant.
Stir in the heavy cream and dried thyme. Bring the mixture to a gentle simmer and cook for 3-4 minutes until the sauce has thickened slightly.
Drain the cooked potatoes and return them to the pot. Add the remaining 3 tablespoons of butter and the milk. Mash until smooth and creamy.
Return the chicken breasts to the skillet, spooning the mushroom cream sauce over them to coat and reheat.
Serve the chicken and mushroom sauce immediately over a portion of mashed potatoes.