Preheat oven to 375 F. Prepare gnocchi according to package directions. Drain and place gnocchi in a single-layer in a 1-½ quart shallow baking dish that has been sprayed with nonstick spray.
Melt butter in a medium saucepan over medium heat. Stir in garlic and cook until fragrant, about 30 seconds. Whisk in flour until it thickens and bubbles, then whisk in milk and Dijon. Continue to whisk mixture and cook until slightly thickened, about 3-5 minutes.
Combine Gruyere and fontina, then add by the handful to milk mixture, stirring until melted before adding the next handful. Once all cheese is melted, season sauce with salt and pepper.
Pour sauce over gnocchi and sprinkle with Parmigiano-Reggiano over top. Bake gnocchi until they puff and cheese is golden and bubbly, about 25 minutes. Let gnocchi rest for 5 minutes before serving.