Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish.
Season chicken breasts with salt and pepper. Heat a large skillet over medium heat; pan-fry chicken until no longer pink in the center, about 10 minutes per side. Shred chicken.
Spread chicken over crescent roll triangles. Roll triangles, starting with the wider end, around chicken. Arrange rolls on prepared baking dish with the seam sides down. Place rolls into prepared baking dish.
Mix cream of mushroom soup, cream of chicken soup, milk, and Cheddar cheese in a bowl; pour mixture over crescent rolls.
Bake in preheated oven until bubbly and rolls are golden brown, about 40 minutes.