Heat 2 Tbs. butter over medium-high heat in a large skillet.
Add onion; cook until soft and golden brown, about 5 minutes. Set aside.
Mix 1 1/2 Tbs. salt and 2 quarts very hot tap water in a 13-by-9-inch lasagna pan.
Add noodles and soak until soft and pliable, 10 minutes.
Drain and stack loosely. (Noodles tend to stick together as they dry, but pull apart easily.).
Make Creamy Parmesan Sauce: Microwave milk, broth and garlic in a 1 quart microwave-safe container on high power until steaming hot, about 8 minutes. (Or slowly heat in a medium saucepan over medium-low heat.)
Fish out garlic; discard.
Melt 2 1/2 Tbs. butter over medium heat in a large saucepan; when foaming subsides, whisk in flour and continue to cook, whisking constantly, until well blended.
Pour in milk mixture all at once and whisk vigorously until sauce is smooth and starts to bubble and thicken.
Stir in 1/2 cup Parmesan, 1/4 teaspoons salt and pepper to taste.
Remove from heat; place plastic wrap directly on the sauce's surface.
Adjust rack to upper-middle position and heat oven to 425 degrees.
Smear 1/4 cup of the sauce over bottom of baking dish.
Assemble the next 4 layers in the following order: 3 noodles, 2/3 cup sauce, 1/4 of the onions, 1/2 of one of the cooked vegetables (alternating each layer), 1/2 cup fontina and 2 Tbs. Parmesan.
Assemble final layer with remaining noodles, sauce and cheeses.
Seal with foil and bake until bubbly, about 35 minutes.
Remove foil, then broil until cheese is spotty brown, 5-7 minutes longer.