Place the grated cucumber on a heavy-duty paper towel or clean kitchen towel. Top with another towel, and pat dry; set aside. Scoop the yogurt into a large square of fine cheesecloth, twist the ends to close, and squeeze to remove excess moisture.
Transfer the strained yogurt to a bowl, and stir in 2 1/2 teaspoons of lemon juice. Gradually add 1 tablespoon olive oil, whisking constantly, until fully incorporated. Mix in the garlic, dill, and drained cucumbers; cover and refrigerate.
Combine the red onion, 1 teaspoon lemon juice, and sumac. Cover and refrigerate.
Place the ground beef in a bowl and add the minced onion, red pepper flakes, coriander, cumin, Urfa biber, and butter. Mix gently with your hands until all of the ingredients are well distributed.
Place bell pepper in blender and blend until it is mostly pureed. Pour off any excess liquid, and add the bell pepper and kosher salt to the meat; mix well. Divide the meat into 4 equal portions and form into patties slightly larger than the buns, as the patties will shrink a bit.
Preheat an outdoor grill for high heat, and lightly oil the grate.
Brush the patties with 2 tablespoons olive oil and grill on high and direct heat about 5 minutes per side or until done to your liking. Place each burger on a bun; top the burgers with cucumber and yogurt tzatziki and the lemon-soaked red onion and sumac mixture.