Preheat the oven to 375°F (190°C).
In a large skillet over medium heat, melt the butter.
Add the chopped onion and minced garlic to the skillet and sauté until the onion is translucent, about 4 minutes.
Sprinkle the flour over the onions and garlic and stir constantly for 2 minutes to form a roux.
Gradually whisk in the turkey or chicken broth and milk, continuing to whisk until the mixture thickens and becomes smooth, about 5 minutes.
Add the dried thyme, dried sage, salt, and black pepper to the sauce and stir to combine.
Add the cooked turkey, frozen peas, carrots, corn, and green beans to the sauce. Stir well to combine all ingredients evenly.
Transfer the turkey and vegetable mixture into a 9×13 inch baking dish and spread evenly.
Sprinkle the shredded cheddar cheese evenly over the casserole mixture.
Open the can of refrigerated biscuit dough and separate the biscuits.
Place the biscuits evenly spaced on top of the casserole, slightly pressing them down.
Bake uncovered in the preheated oven for 30 to 35 minutes, or until the biscuits are golden brown and the casserole is bubbly.
Remove from the oven and let cool for 5 minutes before serving.