Place a rimmed sheet pan on the center rack and preheat the oven to 450°F (232°C).
Add the potatoes to a pot and cover with cold water by about 1 inch. Bring to a boil, season the water generously with salt, then reduce to a lively simmer. Cook 8–10 minutes, just until a knife meets slight resistance in the center. Drain well and shake the colander a few times to rough up the surfaces.
In a large bowl, toss hot potatoes with 2 tablespoons olive oil, 1 teaspoon kosher salt, and 1/2 teaspoon black pepper. Carefully spread onto the preheated sheet pan in a single layer. Roast 15–18 minutes, flipping once, until deeply golden and crisp on the edges.
While the potatoes roast, warm the remaining 1 tablespoon olive oil in a large skillet over medium-low heat. Add the sliced garlic and crushed red pepper; cook 60–90 seconds until fragrant but not browned. Stir in the thyme, lemon zest, and 1 tablespoon lemon juice; remove from the heat to prevent the garlic from overcooking.
Transfer the hot roasted potatoes to the skillet and return heat to medium-high. Toss 1–2 minutes to coat and let the lemon-garlic mixture cling. Sprinkle the Pecorino over the potatoes and toss just until it melts in spots and forms a light, savory crust. Taste and adjust salt if needed; serve immediately.