To prepare vegetables, place a small metal roasting pan in oven. Preheat oven to 450°.

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.

Preheat broiler to high.

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.

Reduce oven temperature to 350°.

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.

Spread 1/2 cup sauce over the bottom of a broiler-safe 13 x 9–inch 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.

Preheat broiler to high.

Broil lasagna for 4 minutes or until top is lightly browned. Sprinkle with 2 tablespoons fresh basil. Let stand for 12 minutes.

This was delicious but here are the changes I will make next time:

The sauce needed further thickening so I used cornstarch dissolved in water. Easier to do this to begin with instead of using flour.

I would shred a rotisserie chicken and layer with the vegetables and sauce.

I would NOT put the sauce in the bottom of the dish and layer as follows:

1/3 lasagna noodles

1/2 squash mixture

1/2 Parmigiano-Reggiano cheese

1/2 chicken

1/3 sauce

REPEAT

Top with last 1/3 of noodles, sauce and fontina or other white cheese.