Place a large pot of water over high heat to boil. Once boiling, salt the water and drop the pasta inches Cook to al dente according to package directions. Drain cooked pasta and reserve.
While the pasta is cooking, place a large skillet over medium heat with the butter. Cook the garlic in the butter until aromatic, about 1 minute, then sprinkle the flour over the butter and cook for another minute. Whisk the milk into the roux and bring up to a bubble to thicken. Add nutmeg, spinach, artichoke hearts, salt and pepper, and cook until the veggies are heated through.
Add the reserved tortellini to the skillet and toss to coat. Transfer everything to a large casserole dish and sprinkle with the cheeses. Place under the broiler to brown, about a minute.