Cut steak into pieces that fit your skillet and season with 1/2 teaspoon salt and pepper. Toss onion rings with buttermilk and let stand for 5 minutes.

Mix flour, cornstarch and remaining salt in a shallow bowl. Heat the frying oil in a deep 10-inch skillet to 350°F, keeping it no more than halfway up the pan.

Lift onions from the buttermilk, dredge lightly and fry in three uncrowded batches for 2 to 3 minutes each until pale golden and crisp. Drain on a rack over a tray.

Warm milk in a small saucepan over low heat. Stir in white American a little at a time until smooth, then keep warm over the lowest heat, stirring occasionally without letting it boil.

Heat a separate heavy skillet over medium-high heat with 1 tablespoon oil. Sear steak for 3 to 5 minutes per side, depending on thickness, until it reaches 145°F.

Rest steak for at least 3 minutes, then slice thinly across the grain. Toast only the cut faces of the rolls under the broiler for 1 to 2 minutes.

Divide steak among the rolls, spoon over the warm cheese sauce and add the fried onion rings. Close gently and serve immediately, one sub per person.