Preheat the oven to 425 degrees.
Cook the pasta according to the directions on the box. Drain, reserving about a cup of the hot pasta water.
While the pasta is cooking, dice the chicken breast into bite-size pieces and mince the garlic cloves.
Peel and wash the carrots and cut them into smaller pieces/thick sticks if they are large.
Spread the cut carrots on a cookie sheet (line it with parchment paper for easier clean up!). Toss the carrots with 2 Tablespoons olive oil and sprinkle with salt and pepper. Bake until browned and tender — about 20 minutes.
In a large skillet, fry the diced chicken in 2 Tablespoons of olive oil until cooked through. Remove the chicken from the pan and set it aside.
In the same skillet, melt the 4 Tablespoons butter over medium-low heat. Add the minced garlic and sauté for a minute.
Whisk in the cream cheese until melted. Whisk in 1/2 cup milk and 1/2 cup chicken broth and stir until smooth.
Stir in the 1 1/2 cups Parmesan cheese and cook until smooth, about a minute. Add salt and pepper to taste.
Stir in the pasta until well-mixed. Add reserved pasta water as needed to thin the sauce.
Top with the chicken. Garnish with fresh herbs if desired. Serve with the roasted carrots.