Heat the oven to 425°F. Toss zucchini with 1 tablespoon oil and 1/4 teaspoon salt, spread on a sheet pan and roast for 15 to 18 minutes until softened; blot excess moisture. Lower oven to 375°F.
Meanwhile, cook lasagna noodles in water with the tablespoon salt until 2 minutes short of the package time and drain. Season chicken with remaining salt and pepper and cook in remaining oil in a skillet for 5 to 7 minutes until at least 165°F.
Melt butter in a saucepan, whisk in flour for 1 minute and gradually add milk and cream. Simmer for 4 minutes, then remove from heat and stir in Parmesan. Mix ricotta and egg in a bowl.
Spread 3/4 cup sauce in a 9-by-13-inch baking dish. Layer three noodles, half the ricotta mixture, half the chicken, half the zucchini, one-third of the mozzarella and 1 cup sauce; repeat, then finish with remaining noodles, sauce and mozzarella.
Cover and bake for 35 minutes, then uncover for 10 to 12 minutes until bubbling and the center reaches 165°F. Rest for 15 minutes before cutting into eight portions.