Prepare the potatoes: Place peeled, cubed potatoes in a large pot of salted water. Bring to a boil and cook for 15-20 minutes until fork-tender. Drain, return to the pot, and mash with 2 tablespoons of butter and warm milk until smooth. Cover to keep warm.
Sear the chicken: Season chicken breasts with salt and pepper. Heat olive oil in a large skillet over medium-high heat. Sear chicken for 5-7 minutes per side until golden brown and cooked through (internal temp 165°F). Remove chicken to a plate and cover.
Cook the carrots: While chicken cooks, place carrots in a separate saucepan with 1/2 cup water, 1 tablespoon butter, and honey. Bring to a simmer, cover, and cook for 10 minutes. Uncover and increase heat to reduce liquid until carrots are glazed and tender.
Make the gravy: In the chicken skillet, melt 1 tablespoon of butter. Add minced garlic, thyme, and rosemary; sauté for 1 minute until fragrant. Pour in chicken broth and scrape up browned bits from the bottom of the pan.
Simmer sauce: Let the broth reduce by half (about 3-4 minutes). Stir in the heavy cream and simmer for another 2-3 minutes until slightly thickened. Taste and adjust salt and pepper.
Serve: Return chicken to the skillet to warm through in the sauce. Serve chicken topped with generous amounts of gravy, alongside the mashed potatoes and glazed carrots.