Preheat oven to 350 degrees F. Place each chicken breast half between two pieces of plastic wrap. Using the flat side of a meat mallet, pound chicken lightly until about 1/8 inch thick. Remove plastic wrap.
For sauce, in a medium saucepan, melt the 3 tablespoons butter over medium heat. Stir in flour. Gradually stir in milk, broth, and wine. Cook and stir until thickened and bubbly. Set aside.
For filling, in a medium skillet, heat the 1 tablespoon butter over medium heat. Add mushrooms and onion; cook until tender. Stir in crabmeat, crackers, parsley, salt, and pepper. Stir in 2 tablespoons of the sauce. Place about 1/4 cup of the filling on each piece of chicken. Fold in sides; roll up. Secure with wooden toothpicks, if necessary. In a large skillet, heat the 2 tablespoons butter over medium heat. Cook chicken rolls, half at a time, in hot butter until brown on all sides.
Arrange chicken rolls, seam sides down, in an ungreased 3-quart rectangular baking dish. Top with the remaining sauce. Bake, covered, about 35 minutes or until chicken is no longer pink. Sprinkle with cheese and paprika. Bake, uncovered, about 2 minutes more or until cheese is melted. Remove toothpicks, if using.
Transfer chicken to a serving platter. Whisk mixture in baking dish; pass with chicken.