Heat a large skillet over medium heat with a drizzle of oil. Add onions and a pinch of salt; cook 20–25 minutes, stirring occasionally, until golden. Add mushrooms and cook about 10 minutes until moisture releases and mushrooms brown. Add brown sugar, Worcestershire, smoked paprika, rosemary, and thyme; cook another 5–10 minutes until jammy. Set aside.

Whisk mayonnaise, Worcestershire, Dijon, hot honey, truffle oil, smoked paprika, and AP seasoning together. Taste and refrigerate until needed.

Form 8-ounce beef portions into patties slightly wider than the buns, with a small indent in the center. Season both sides with kosher salt, pepper, and AP seasoning.

Heat a cast-iron skillet over high heat until very hot. Add a little oil and sear patties without moving for 3–4 minutes. Flip once. Add butter, smashed garlic, rosemary, and thyme; spoon foaming herb butter over patties for 1–2 minutes. Add both cheddars during the final minute and cover loosely to melt. Cook beef to your preferred safe doneness.

Toast brioche buns cut-side down in the herb butter for 60–90 seconds until golden.

Spread honey truffle sauce on both buns. Add patty and mushroom-onion jam. Top with bun and serve immediately.