Preheat Blackstone to 400-425°F (8-10 minutes). Lightly oil the surface.
Form the beef into 8 equal 2-oz balls. Don't compress.
Melt 2 tbsp butter on a cooler zone of the griddle. Add mushrooms in a single layer. Brown undisturbed 2-3 minutes.
Add garlic and thyme to mushrooms; toss 30 seconds. Add Worcestershire and reduce 30-45 seconds. Salt to taste. Push to a holding zone and dot with remaining 1 tbsp butter.
Place meatballs on hot zone, spaced 3 inches apart. Cover each with parchment square and smash hard to ¼ inch thick. Season tops generously.
Cook undisturbed 90 seconds to 2 minutes, until edges are deeply crisp and top has sweated.
Flip with thin spatula, scraping under crust. Top each patty with American cheese; stack a second patty on top of every other; finish with Swiss.
Splash water beside the patties, cover with basting dome 60 seconds to melt cheese.
Spread mayo on cut sides of buns; toast face-down on the griddle 60-90 seconds until golden.
Assemble: bottom bun, double-patty stack, generous spoonful of mushrooms, lettuce/onion if using, top bun. Serve immediately.