Prepare a grill or grill pan for medium-high heat.
For the salad: Mix the sliced cucumbers with the garlic, Greek yogurt, dill, mint and lemon juice. Season to taste with salt and pepper.
For the burger: In a bowl, mix together the ground beef, oregano and garlic. Form into four 6-ounce patties. Grill for 3 to 4 minutes per side, or to desired doneness.
Spread the burger buns with mayonnaise. Layer the bun bottoms with a burger patty, a roasted tomato, and then some red onions and feta. Cover with the bun tops. Serve with cucumber salad on the side.
Preheat the oven to 350 degrees F.
Slice the tomatoes in half through the meridian and place cut-side up on a baking sheet. Drizzle with olive oil and sprinkle with salt and oregano. Roast until the tomatoes are beginning to collapse and have browned a bit on top, about 2 hours.