Preheat the oven to 350°F and grease a 9x13-inch baking dish. Set the cream cheese out to soften if you haven't already.
Bring a large pot of salted water to a boil. Cook the egg noodles for 1 minute less than the package instructions state. Drain and set aside.
In a large skillet, melt 2 tablespoons of butter over medium heat. Add the onion and cook for 4–5 minutes until soft. Add the garlic and cook for 30 seconds more, until fragrant.
Reduce heat to low. Whisk in the cream of chicken soup, sour cream, softened cream cheese, and chicken broth until smooth and glossy, about 2 minutes.
Stir in the shredded chicken, peas and carrots, 1 cup of the cheddar, salt, and pepper. Fold in the cooked noodles gently until evenly coated.
Transfer the mixture to the prepared baking dish and spread into an even layer.
In a small bowl, toss the crushed crackers with the melted butter. Sprinkle over the casserole along with the remaining 1 cup of cheddar.
Bake uncovered for 25–30 minutes, until the topping is golden and the sauce is bubbling gently at the edges.
Let rest for 5 minutes before serving.