Position racks in upper and lower third of oven; preheat to 425 degrees F.

Toss potatoes with 1 tablespoon oil in a large bowl until well coated. Divide between 2 large baking sheets and spread in an even layer. Roast, rotating the pans from top to bottom about halfway through, until tender and beginning to brown, about 20 to 25 minutes.

Meanwhile, heat the remaining 2 tablespoons oil in a large saucepan over medium heat. Add onion; cook, stirring frequently, until very soft and golden brown, 5 to 8 minutes. Add garlic; cook, stirring, until fragrant, about 1 minute more.

Add flour, salt, and pepper; cook, stirring for 1 minute more. Add milk; cook, stirring and scraping up any brown bits at the bottom of the pan. Increase heat to medium high; cook, stirring, until the sauce thickens and bubbles, 3 to 5 minutes. Remove from heat. Squeeze spinach to remove any excess moisture; stir into the sauce.

When the potatoes are done, remove them from the oven. Preheat the broiler. Transfer half the potatoes to a 2-quart broiler-safe baking dish. Spread half the sauce over the potatoes and top with 1/2 cup cheese. Add the remaining potatoes and top with the remaining sauce and cheese.

Broil, watching carefully, until the sauce is bubbling and the cheese is beginning to brown, 1 to 5 minutes, depending on your broiler. Let stand 10 minutes before serving.