Step 1

Line a rimmed baking sheet with aluminum foil and place on the center rack of the oven. Preheat the oven to 425 degrees F (see Cook's Note).

Step 2

For the Parmesan potatoes: Toss the potatoes, thyme sprigs, olive oil, 1 teaspoon salt and several grinds of black pepper in a large bowl until combined. Carefully remove the hot baking sheet from the oven and spray with nonstick cooking spray. Lay the potatoes cut-side-down in an even layer. Bake until tender and golden brown, about 35 minutes.

Step 3

For the smoked paprika aioli: Meanwhile, whisk together the mayonnaise, lemon juice, Dijon, paprika, garlic, a pinch of salt and a couple grinds of black pepper in a small bowl until smooth. Cover with plastic wrap and refrigerate until ready to serve.

Step 4

Remove the baking sheet from the oven and push the potatoes to one side of the pan (try to give yourself as much empty surface area as possible). Lightly spray the foil with more cooking spray, then sprinkle the Parmesan in an even layer over the empty space on the baking sheet, making sure to go all the way to the edges to create a roughly 14-by-12-inch rectangle. Place the potatoes on top of the Parmesan layer, cut-side down, so that all the potatoes are sitting on the cheese.

Step 5

Return the baking sheet to the oven and bake until all the Parmesan is golden brown and crisp, 8 to 10 minutes more. Cool slightly, about 5 minutes, then use a spatula to lift the crispy, cheesy potatoes off the pan (some potatoes will come off individually while others will come off in pairs). Serve with the smoked paprika aioli on the side.