Place milk in one bowl and bread crumbs and 1/4 cup parmesan cheese in a separate shallow bowl.
Dip chicken in milk, then coat with crumbs.
In a skillet over medium heat, melt butter. When butter is completely melted and hot, add the chicken. Cook chicken in butter until browned on both sides and the internal temperature reaches at least 165 degrees. Remove chicken from skillet and keep in a 9x13 pan in the oven on warm.
Add chicken broth to skillet and stir to loosen up browned bits from pan. Stir in cream, basil, and salt and pepper and bring to a boil. Turn heat to low and whisk in remaining parmesan cheese.
When the sauce has thickened to your desired consistency, pour sauce over the chicken and serve.