Remove all fat from chicken breasts and cut into 1 inch by 4 inch strips. In a bowl, whisk together the eggs and buttermilk. In a plastic resealable bag mix together the bread crumbs, grated cheese, garlic powder, salt, pepper and dill weed.
Dip each chicken strips into buttermilk mixture, then into bread crumb mixture (about 4 strips at a time), and toss to coat well. Place strips in a single layer in a 9x13 inch baking dish and drizzle with melted butter/margarine. Bake in the preheated oven for 15 minutes or until golden brown, turning once after about 8 minutes of baking time.