Fill large bowl with ice water. Bring 4 quarts water to boil in large Dutch oven or stockpot over high heat; add salt and spinach, stirring until spinach is just wilted, about 5 seconds. Using skimmer or fine-mesh strainer, transfer spinach to ice water and let stand until completely cool, about 1 minute, then drain spinach and transfer to clean kitchen towel. Wrap towel tightly around spinach to form ball and wring until dry. Chop spinach medium and set aside.

Melt 5 tbsp butter in medium saucepan over medium heat; add minced onions and garlic and cook, stirring frequently, until onions have softened, about 4 minutes. Add flour and cook, stirring constantly, for about 1 ½ minutes. Gradually whisk in milk. Bring mixture to boil over medium-high heat, whisk in bay leaves, nutmeg, ½ teaspoon salt, and pepper. Reduce heat to low and simmer 10 minutes, whisking occasionally. Discard bay leaves and whisk in Parmesan. Measure out and reserve ½ cup sauce in a small bowl; pressing plastic wrap directly against the surface. Add the chopped spinach to the remaining sauce and stir until combined. Transfer spinach sauce to bowl, press plastic wrap directly against surface, and set aside.

Blend cottage cheese, egg, and salt in food processor or blender until very smooth, about 30 seconds. Adjust oven rack to middle position and heat oven to 425 degrees. Place noodles in 13- by 9-inch broiler-safe baking dish and cover with hot tap water; let soak 5 minutes, stirring the water and noodles occasionally to prevent sticking. Remove noodles from water and place in single layer on kitchen towel. Wipe baking dish dry and spray with non-stick cooking spray.

Use rubber spatula to distribute the reserved ½ cup béchamel in bottom of baking dish; position 3 noodles on top of sauce. Spread 1 cup spinach mixture evenly over noodles, sprinkle evenly with Parmesan, and top with 3 more noodles. Spread 1 cup spinach mixture evenly over noodles, sprinkle evenly with 1 cup fontina, and top with 3 more noodles. Spread 1 cup spinach mixture evenly over noodles, followed by cottage cheese mixture. Finish with 3 noodles, remaining cup spinach mixture, and remaining cup of fontina. Lightly spray large foil sheet with cooking spray and cover the lasagna. Bake until bubbling, about 20 minutes, then remove the foil. Remove the pan and adjust oven rack to uppermost position (about 6 inches from heating element) and heat broiler. Broil lasagna until cheese is spotty brown, 4 to 6 minutes. Cool 10 minutes, then cut into pieces and serve.