Heat the grill to high.
Reserve 2 tablespoons each cucumber and tomato and the scallion greens for garnish. Mix the yogurt, feta, chopped jalapenos, remaining cucumber and tomatoes, and scallion whites in a medium bowl. Season to taste with salt and pepper. Refrigerate until ready to serve.
Mix the paprika, cumin, brown sugar, and 1 teaspoon each salt and pepper in a large bowl. Add the beef and mix with your hands until evenly spiced. Form the mixture into 4 1/2-inch-thick patties.
Grill until the meat is browned and cooked through, about 4 minutes per side. Transfer to a cutting board and let rest while you grill the pitas, about 1 minute per side.
Divide the grilled pitas among serving plates. Finely chop the meat and scatter over the pitas, then top with the yogurt sauce. Garnish with cucumber, tomatoes, scallion greens, and jalapeno slices, and serve.