Stir in the mushrooms and garlic. Cook, stirring occasionally, until the mushrooms release their moisture and then brown and reduce, about 6–8 minutes. The pan should look dry and the mushrooms deeply caramelized.
Season with the Worcestershire sauce, salt, and pepper. Cook 1–2 minutes more, until flavors are well blended. Transfer the mushroom mixture to a bowl and let cool slightly.
If using breadcrumbs or oats, mix them with the mushroom filling along with the egg (or flax egg) to help bind. The mixture should come together when pressed; if it’s too wet, add a little more breadcrumbs.
Form two patties that fit the buns, about 1/2 inch thick. Lightly press to compact, then refrigerate for 10 minutes to help them hold shape while you toast the buns.
In the same skillet, toast the cut sides of the buns in a touch more butter until golden. Remove and set aside. Place one cheese slice on the bottom half of each bun.
Cook the patties in the skillet over medium heat, about 4–5 minutes per side, until deeply browned and cooked through. Lay a patty on each cheese-topped bun half, pile the mushroom mixture on top, add a second cheese slice, and crown with the top bun.
Press gently with a spatula to help the cheese melt into a cohesive melt. If needed, cover the pan for a minute to trap heat and finish melting. Slice and serve immediately.
