Gently mix the turkey, salt, paprika, and half of the beaten egg in a large bowl just until blended.
Using a 4-inch-diameter ring mold, form the turkey mixture into four 8-ounce patties on a baking sheet lined with waxed paper.
Cover and refrigerate for at least 1 hour for flavors to blend. Flatten the patties so they are slightly larger than the diameter of the buns.
To make the relish: Heat the olive oil in a large sauté pan over medium-high heat. Add the onion and sauté until translucent, about 3 minutes. Add the bell peppers, sugar, garlic, and thyme and cook until the vegetables soften slightly, about 3 minutes. Add the vinegar and cook until the liquid has evaporated, about 2 minutes. Remove the skillet from the heat. Stir in the parsley and season to taste with salt and pepper; set aside to keep warm.
Preheat a grill over medium-high heat. Lightly brush the patties with olive oil and season with salt and pepper, then place the patties on the grill. Cook until fully cooked through and an instant-read thermometer registers 165°F when inserted into the center of the patties, about 4 minutes per side. While the burgers are cooking, brush the cut sides of the buns lightly with olive oil and grill, oiled side down, until lightly toasted with grill marks, about 1 minute. Remove the buns and burgers from the grill and allow the burgers to rest for 3 minutes.
To make the yogurt sauce: Mix the yogurt, olives, shallots, and parsley in a medium bowl. Season the yogurt sauce to taste with salt and pepper.
Set a patty atop each bun bottom. Top with the warm red pepper relish, then yogurt sauce, and finally with the bun tops.