Add the sliced red onion for the cucumber salad to a medium bowl with the lemon juice and ¼ teaspoon salt. Toss and let sit for at least 10 minutes to soften.
Place the grated onion for the patties in a clean kitchen towel or a few layers of paper towel and squeeze out as much excess liquid as possible.
Add the squeezed onion to a large bowl along with the beef, garlic, parsley, mint (if using), cumin, salt, smoked paprika, and black pepper. Mix with your hands until evenly combined.
Divide the mixture into 4 portions and shape into sturdy oval patties about ¾-inch thick. Lightly brush both sides with olive oil. Refrigerate for 15 minutes if the mixture feels especially soft.
Preheat the grill to medium-high. Clean and oil the grates well.
While the grill heats, to the bowl with the onion, add the cucumber and remaining ¼ teaspoon salt and toss to coat. Then toss in the parsley, and mint. Adjust the lemon juice and salt to taste. Drizzle with olive oil and toss once more.
In a small bowl, stir together the yogurt, lemon juice, and salt until smooth. Add a splash of water if you prefer a thinner, drizzleable sauce.
Grill the beef patties for 4–5 minutes per side, or until nicely browned and the centers reach 160°F.
Serve the patties with the cucumber salad, lemony yogurt, and warm pita or starch of choice.