Preheat the broiler. Line a rimmed baking sheet with aluminum foil.
Melt the butter in a medium skillet over medium-low heat. Add the sliced onions, 1 tablespoon fresh thyme or 1 1/2 teaspoons dried, 1/2 teaspoon kosher salt and a few grinds of pepper. Cook, stirring occasionally, until the onions are deep golden brown and caramelized, about 15 minutes. If the skillet gets too brown, add 2 tablespoons water and scrape the brown bits from the bottom of the skillet with a wooden spoon; continue to cook until all the water has evaporated. Add 1 tablespoon of the Worcestershire and stir until completely absorbed, about 1 minute more.
Meanwhile, place the muffins or buns on the prepared baking sheet and broil until toasted and browned along the edges, 1 to 2 minutes. Set aside for serving.
Put the mushroom caps, gill-side up, on the same baking sheet. Broil until they release their juices and soften slightly, about 6 minutes. Flip the mushrooms, brush the tops with the oil and season with the garlic salt. Broil until lightly browned and the juices have cooked off, about 6 minutes more. Top each mushroom with 1/4 cup cheese, then return to the broiler until the cheese melts, 1 to 1 1/2 minutes.
While the mushrooms are broiling, stir together the mayonnaise, garlic, remaining 1 teaspoon fresh thyme or 1/2 teaspoon dried thyme and 1 teaspoon Worcestershire in a small bowl until smooth. Season with a pinch of kosher salt and pepper. Spread on the inside of the toasted muffin or bun tops. Place 1 portobello mushroom on each muffin or bun bottom. Layer with the caramelized onions and crispy fried onions, then close the burgers and serve.