1. To prepare vegetables, place a small metal roasting pan in oven. Preheat oven to 450°.
2. Combine shallots and squash; drizzle with 2 tablespoons oil. Sprinkle with 1/2 teaspoon salt and 1/2 teaspoon pepper; toss. Arrange vegetables in preheated pan; roast at 450° for 30 minutes, stirring once.
3. Preheat broiler to high.
4. Place tomatoes in a roasting pan; broil 10 minutes or until blistered. Combine squash mixture, tomatoes, 1/4 cup basil, chives, and garlic; toss gently.
5. Reduce oven temperature to 350°.
6. To prepare sauce, heat a small saucepan over medium heat. Add remaining 1 tablespoon oil; swirl. Combine broth and flour, stirring with a whisk; add broth mixture to oil. Bring to a simmer, stirring constantly with a whisk. Stir in cream, mustard, 1/4 teaspoon salt, and 1/4 teaspoon pepper; simmer gently 4 minutes or until mixture begins to thicken, stirring constantly. Remove from heat; let stand 5 minutes. Place egg in a small bowl; slowly add 1 1/2 cups sauce to egg, stirring constantly. Return sauce to pan.
7. Spread 1/2 cup sauce over the bottom of a broiler-safe 13 x 9inch ceramic baking dish coated with cooking spray. Arrange 3 lasagna noodles over sauce; spread half of squash mixture over noodles. Sprinkle with half of the Parmigiano-Reggiano cheese. Repeat the layers with remaining noodles, vegetable mixture, and Parmigiano-Reggiano, ending with noodles. Pour remaining sauce over noodles. Sprinkle with fontina. Bake at 350° for 30 minutes or until bubbly.
8. Preheat broiler to high.
9. Broil lasagna for 4 minutes or until top is lightly browned. Sprinkle with 2 tablespoons fresh basil. Let stand for 12 minutes.