Preheat the oven to 350°F (175°C). Grease a 9×13-inch casserole dish with butter or cooking spray.
Cook the egg noodles according to package instructions until al dente. Drain and set aside.
In a large skillet, heat the olive oil over medium heat. Add the chopped onion and cook until softened, about 4 minutes.
Add the minced garlic and cook for an additional 1 minute until fragrant. Remove from heat.
In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes until it forms a smooth paste and starts to bubble.
Gradually whisk in the milk, stirring constantly to avoid lumps. Continue cooking and stirring until the sauce thickens, about 5-7 minutes.
Remove the sauce from heat and stir in the sour cream, salt, black pepper, paprika, and dried thyme.
Add the shredded cheddar cheese, mozzarella cheese, and Parmesan cheese to the sauce. Stir until the cheese is completely melted and the sauce is smooth.
In a large mixing bowl, combine the cooked egg noodles, shredded chicken, sautéed onion and garlic, thawed peas, and cheese sauce. Mix well to combine.
Transfer the mixture to the prepared casserole dish and spread evenly.
Sprinkle the crushed buttery crackers evenly over the top of the casserole.
Bake in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the topping is golden brown.
Remove from the oven and let it cool for 5 minutes before serving.