Make the Sauce: In a small bowl, combine Greek yogurt (or mayonnaise), lemon juice, Dijon mustard, honey (or maple syrup), and minced garlic. Season with salt and pepper to taste. Adjust seasoning or sweetness as needed, then chill while you prepare the burgers.
Season the Halloumi: In a shallow dish, whisk olive oil with oregano, paprika, salt, and pepper. Coat each halloumi slice evenly in the mixture.
Cook the Halloumi: Heat a grill pan or non-stick skillet over medium heat. Cook the halloumi for 2-3 minutes per side until golden brown and crisp on the edges. Avoid overcooking to keep it tender inside.
Toast the Buns: Brush the cut sides of the brioche buns with olive oil. Toast them in the same pan, cut side down, until golden and slightly crisp.
Assemble the Burgers: Spread a layer of sauce on the bottom bun. Add lettuce, then stack the grilled halloumi, followed by tomato, red onion, and any optional toppings. Spread more sauce on the top bun and close the burger. Enjoy warm with sweet potato fries, chips, or a side salad.