Preheat the oven to 400°F (200°C).
In a large bowl, combine ground turkey, chopped onion, minced garlic, breadcrumbs, egg, salt, black pepper, dried thyme, and dried sage. Mix well until all ingredients are evenly incorporated.
Form the turkey mixture into 4 equal-sized patties.
Heat 1 tablespoon of olive oil in a large skillet over medium heat. Add the turkey patties and cook for 5 minutes on each side or until browned and cooked through. Remove patties from skillet and set aside.
In a large bowl, toss carrots, potatoes, and green beans with the remaining 1 tablespoon of olive oil, salt, and pepper to taste.
Spread the vegetables evenly on a baking sheet and roast in the preheated oven for 25-30 minutes, or until tender and slightly caramelized, stirring halfway through.
While the vegetables roast, prepare the gravy. In the same skillet used for the turkey patties, melt butter over medium heat.
Whisk in flour and cook for 1-2 minutes until lightly golden.
Gradually whisk in chicken broth, stirring constantly to avoid lumps.
Bring the mixture to a simmer and cook until thickened, about 5 minutes.
Season the gravy with salt and black pepper to taste.
Serve the turkey patties hot with roasted vegetables on the side, topped with warm gravy.