Preheat oven to 375°F and lightly grease a 9x13 inch baking dish.
Cook egg noodles according to package directions; drain and set aside.
In a large skillet over medium heat, melt 1 tablespoon unsalted butter.
Add chopped yellow onion and cook until softened and translucent, about 5 minutes.
Add minced garlic and cook 1 minute more, until fragrant.
In a large bowl, whisk together cream of chicken soup, sour cream, and chicken broth until smooth.
Stir in dried thyme, black pepper, and a pinch of salt.
Add cooked shredded chicken, sautéed onion and garlic, frozen mixed vegetables, cooked egg noodles, and shredded sharp cheddar cheese to the soup mixture; stir until combined.
Pour mixture into prepared baking dish and spread evenly.
In a small bowl, combine crushed buttery crackers and melted unsalted butter; sprinkle evenly over casserole.
Bake for 25–30 minutes, until bubbly at edges and topping is golden brown.
Let rest 5–10 minutes before serving.