Prepare the mashed potatoes by placing the peeled and chopped potatoes into a large pot of salted water and bringing to a boil. Reduce heat and simmer until tender. Drain the water and mash the potatoes, mixing in butter, milk, salt, and pepper until smooth and creamy. Keep warm while preparing the chicken.
Cook the creamy herb chicken by heating olive oil and butter in a skillet over medium heat. Season the chicken with salt, pepper, thyme, and rosemary. Sear until golden brown on both sides and fully cooked, then remove from the pan and set aside. In the same skillet, sauté minced garlic until fragrant, pour in heavy cream and chicken broth, and stir in Parmesan cheese until the sauce thickens. Return the chicken to the skillet, coating it in the sauce, and garnish with parsley.
Prepare the glazed carrots by melting butter in a saucepan over medium heat, adding the sliced carrots, brown sugar, and salt. Cook while stirring occasionally until the carrots are tender and lightly caramelized.
Assemble the meal by placing a scoop of mashed potatoes on each plate, topping with a chicken breast coated in the creamy herb sauce, and arranging the glazed carrots on the side. Drizzle any remaining sauce over the chicken and potatoes for added richness.
