In a large bowl, combine beef, egg, breadcrumbs, garlic powder, onion powder, and paprika; season with salt and pepper. Divide beef mixture into 4 patties, then flatten to ½"-thick oval patties.
In a large skillet over medium heat, heat oil. Add patties and cook, turning halfway through, until browned and cooked through, 6 to 8 minutes per side. Transfer to a plate; keep warm.
In same skillet over medium heat, melt butter. Add mushrooms and onions and cook, stirring occasionally, until softened, 6 to 8 minutes. Add flour and toss to coat veggies. Slowly whisk in broth, then add Worcestershire and thyme. Bring to a boil, then reduce heat to medium low and bring to a simmer. Cook, stirring occasionally, until thickened, about 5 minutes; season with salt and pepper.
Return steaks to skillet and spoon sauce over. Cook until warmed through, about 1 minute more.
