Preheat oven to 400°F (200°C).
Prep potatoes, onions, carrots, and garlic.
In a large bowl, toss potatoes, onions, carrots, and garlic with salt, pepper, dried thyme, dried oregano, and olive oil.
Add sausages and toss to coat lightly.
In a bowl, whisk melted butter and flour until smooth. Slowly whisk in beef broth until smooth.
Transfer vegetables and sausages to a 9x13-inch baking dish. Pour the broth sauce over everything.
Bake 40–50 minutes, turning sausages halfway through, until potatoes are tender and sausages are browned and cooked through.
Rest 5 minutes, then serve with spoonfuls of the pan sauce.