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In a large frying pan, heat half the oil and sautée onions and mushrooms. Add garlic and continue cooking for 1 minute. Transfer to a bowl.
In the same pan, brown ground chicken in the rest of the oil, breaking it up with a fork. Add salt and pepper. Transfer to a bowl.
Add parsley and parmesan to the chicken mixture. Add salt and pepper. Set aside.
Melt butter in the same pan. Add flour and cook 1 minute over medium heat, stirring.
Gradually add milk and bring it to the simmer, stirring constantly with a whisk. Add salt and pepper. Pour onto the chicken mixture.
Preheat oven to 350°F (180°C) and place oven rack in middle position.
Spread a third of the chicken mix in a 13 X 9 in. (33 X 23 cm) baking pan. Cover it with the squash slices and a row of lasagna sheets. Assemble the other two layers, finishing with the chicken mixture. Cover with the cheddar cheese.
Bake about 45 minutes. Finish cooking under the broiler about 5 minutes.
Try this recipe with sautéed mushrooms—morels, oyster mushrooms, shiitake mushrooms—instead of squash.