2/3 cup plain Greek yogurt (with at least 2% milk fat; do not use non-fat)
1/3 cup grated seedless English cucumber, patted dry with paper towels
2 teaspoons chopped fresh dill fronds
1 teaspoon fresh lemon juice
1/4 teaspoon kosher salt
1 small clove garlic, minced
16 ounces ground pork
8 ounces white button mushrooms, finely chopped
1 teaspoon freshly grated lemon zest
1 teaspoon dried oregano
1 teaspoon kosher salt
1/4 teaspoon cracked black pepper
1/4 cup fresh lemon juice
1/4 cup extra-virgin olive oil
4 butter lettuce leaves
4 soft white hamburger buns, split
4 sliced rings red onion
2 tablespoons crumbled feta cheese
16 thin slices seedless English cucumber
8 slices Roma tomato
For the tzatziki sauce: Combine the yogurt, cucumber, dill, lemon juice, salt and garlic in a small bowl. Cover and refrigerate until ready to serve.
For the burgers: Combine the pork, mushrooms, lemon zest, oregano, salt and pepper in a medium bowl. Form into 4 equal-size patties. Pour the lemon juice and olive oil into separate shallow bowls. Dip both sides of each patty into the lemon juice, then into the olive oil to coat.
Heat a large nonstick skillet over medium heat. Cook the patties in the hot skillet, flipping once, until the pork is just cooked through, 4 to 6 minutes per side.
For the assembly: Put 1 lettuce leaf on each bottom bun. Dollop each leaf with about 2 tablespoons tzatziki sauce (reserve the remaining sauce), then top with 1 patty and 1 red onion slice. Mix the feta into the reserved tzatziki sauce and spoon 1 to 2 teaspoons onto each burger, followed by 4 cucumber slices and 2 tomato slices. Close the burgers with the top buns. Serve immediately.