Preheat oven to 400°F. Lightly grease a 13x9-inch baking pan with vegetable cooking spray. Set aside.
Bring 4 quarts water to a boil in a large pot. Add egg noodles and 1 teaspoon salt. Cook, stirring frequently, until almost al dente, about 2 minutes below the recommended cooking time. Drain and rinse with cold water. Pour the noodles back into the pot and add 1 tablespoon butter. Toss to coat. Set aside.
While the noodles are cooking, warm a large skillet over medium-high heat. Add olive oil and 1 tablespoon butter. Once melted add the sliced mushrooms and cook, undisturbed, until the mushrooms are very dark on one side. Stir and turn the mushrooms and continue cooking until the moisture is released then evaporated, and the mushrooms are well browned. Transfer the mushrooms to the pot with the noodles.
Melt another tablespoon of butter in the now empty skillet. Add the carrots and onions and cook, stirring occasionally, until softened, about 5 minutes. Transfer all the vegetables to the pot with the noodles.
In the empty skillet melt 3 tablespoons butter. Add the flour and cook, whisking constantly for about 1 minute. Slowly whisk in the milk and broth. Bring to a boil. Reduce the heat to medium-low and simmer until slightly thickened, about 5 minutes. Whisk both cheeses into the sauce until smooth. Add garlic powder, Dijon mustard and 1 teaspoon black pepper and ½ teaspoon salt. Whisk until blended. Pour the sauce over the reserved noodles and vegetables.
Add the diced chicken and peas to the noodle mixture and gently fold with a spatula until combined. Pour the chicken and noodles into the prepared baking pan.
Melt the remaining 2 tablespoons of butter in the microwave. Sprinkle crushed crackers over the casserole, then drizzle with melted butter.
Bake until golden brown and bubbling, about 15 minutes. Tent with foil about halfway through to prevent over-browning. Allow the casserole to cool for 10 minutes before serving.