Add the ground turkey, mayonnaise, panko crumbs, 1/2 teaspoon salt, a few grinds of black pepper and 2 teaspoons of the fresh thyme leaves to a mixing bowl. Using your hands, combine gently but thoroughly and form the mixture into four 1/2-inch thick patties. Transfer to a plate and refrigerate for at least 15 minutes.

While the burgers chill, heat 1 tablespoon of the butter along with the olive oil in a large frying pan over medium heat. Add the onion and sauté until softened, 4 to 5 minutes.

Add the garlic and sauté just until fragrant, 1 minute. Add the mushrooms, season with salt and pepper and continue cooking until the mushrooms are nicely browned, 7 to 8 minutes more.

Add 2 tablespoons of the sherry and continue cooking until the liquid is almost evaporated. Add the remaining teaspoon of thyme, cook 1 minute longer, then transfer the mixture to a bowl and set aside.

In the same pan, heat the remaining tablespoon of butter over medium-high heat, add the turkey burgers and cook until lightly browned, 1-1/2 minutes per side. Add the rest of the sherry and continue cooking the burgers, turning several times, until the sherry has evaporated and the patties are nicely glazed, 3 to 5 minutes.

Reduce the heat to low and spoon a portion of the mushroom mixture onto each burger. Top with a slice of cheese and cover the pan just long enough for the cheese to melt.

Transfer the burgers to the rolls. Add lettuce and tomato and serve immediately.