Cut the potatoes into thick wedges and submerge in cold water for at least 30 minutes.
Whisk the cornstarch, paprika, garlic powder, onion powder, cayenne, salt, and pepper together in a large bowl.
Drain the potatoes and pat completely dry, then toss in the seasoned cornstarch mixture until evenly coated.
Heat oil to 325°F in a large pot and fry the wedges in batches for 4–5 minutes, until pale gold. Transfer to a wire rack.
Increase the oil to 375°F and fry the wedges a second time for 2–3 minutes, until deep golden brown and audibly crisp.
While the second fry finishes, melt the cheese and milk together in a small saucepan over low heat, stirring until smooth. Stir in jalapeños if using.
Sprinkle the hot wedges with extra salt immediately after frying, then serve with the warm cheese sauce for dipping.