Preheat oven to 400 degrees F.
Trim any connective tissue and all fat off ham steak. Mince the meat into very small bits and set aside.
Heat a 12-inch nonstick skillet with oven safe handle over moderate heat. Add oil and 1 tablespoon butter to the skillet and coat sides and bottom of pan evenly with melted butter and oil mixture. Add ham bits to the pan and saute 3 minutes to brown them a bit and render some of the moisture in the meat.
Whisk together eggs and milk or half-and-half. Break off tiny pieces of remaining 1 tablespoon butter and drop them into beaten eggs. Season eggs with a little salt and pepper and whisk again to combine. Pour eggs into skillet over ham. Stir eggs gently to evenly distribute bits of ham throughout the eggs.
As eggs set, lift up bottom skin that has formed and allow uncooked eggs to settle. Keep doing this as eggs brown until the eggs begin to set. Transfer the pan to the oven, and allow the frittata to cook 10 minutes, or until golden on top. Add a generous layer of shredded fontina to the frittata and leave in oven another 3 to 5 minutes, or until cheese is melted and begins to bubble and brown. Serve frittata wedges directly from the skillet with a pie server.