In a large bowl, mix egg, bread crumbs, Worcestershire sauce, garlic powder, onion powder, salt, and pepper. Add beef and mix with clean hands until mixture becomes tacky.

In a large skillet over medium-high heat, heat 1 tsp. oil. Pinch off a small piece of beef mixture and add to pan. Cook until browned on both sides, about 1 minute total. Transfer to a small plate. Let cool slightly, then taste. Season remaining beef mixture with more salt, if needed.

Form beef mixture into 4 (3"-wide, 4"-long, ¾"-thick) oval patties and arrange on a parchment-lined sheet. Using a clean finger, press 8 to 10 dimples into each patty.

In a large skillet over medium-high heat, heat remaining 1 Tbsp. oil. Place patties in skillet, reduce heat to medium, and cook, undisturbed, until deeply browned on the bottom, 2 to 3 minutes. Turn and cook until other side is browned, 2 to 3 minutes more. Transfer to a large plate.

Drain all but 3 Tbsp. fat from skillet. Reduce heat to medium and add mushrooms and ¼ c. water. Cook, stirring occasionally, until deeply golden brown, 6 to 8 minutes; season with ½ tsp. salt. Transfer to a small bowl; set aside.

In same skillet over medium heat, melt butter. Add onion, thyme, and ½ tsp. salt. Cook, stirring occasionally, until onions are slightly softened and translucent, about 7 minutes. Add flour and tomato paste and cook, stirring frequently, until tomato paste is darkened and beginning to stick to skillet, about 2 minutes. Add wine and cook, stirring, until slightly reduced and a thick, jammy paste forms, about 2 minutes. Slowly whisk in stock and Worcestershire sauce. Bring to a boil over high heat, then reduce heat to medium-low and simmer, stirring occasionally, until thickened, 4 to 5 minutes.

Remove thyme sprigs and stir in reserved mushrooms. Return steaks to skillet, spooning sauce over. Simmer until an instant-read thermometer inserted into thickest part of steak registers 165°, 4 to 5 minutes more; season with remaining ½ tsp. salt, if needed.