Preheat oven to 400°F (200°C).
Toss halved potatoes with 2 tablespoons of olive oil, rosemary, salt, and pepper on a baking sheet.
Roast potatoes for 25-30 minutes, turning once, until golden brown and tender.
Season chicken breasts generously with salt and pepper.
Heat 1 tablespoon of olive oil and 1 tablespoon of butter 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 skillet and set aside.
Add the remaining 1 tablespoon of butter and the sliced mushrooms to the skillet; sauté for 5-7 minutes until browned.
Add minced garlic and cook for 60 seconds until fragrant.
Pour in chicken broth to deglaze the pan, scraping the bottom to release browned bits.
Stir in heavy cream and thyme; simmer for 5 minutes until the sauce thickens.
Whisk in Parmesan cheese until melted and smooth.
Return the chicken to the skillet, spooning sauce over it, and serve immediately alongside the roasted potatoes.