Instructions

Adjust 1 oven rack to lowest position and second rack 6 inches from broiler element. Place rimmed baking sheet on lower rack and heat oven to 450 degrees. Combine ½ cup plain Greek yogurt, 3 tablespoons chopped basil, 1 tablespoon sun-dried tomato oil, and ⅛ teaspoon table salt in bowl; season with salt and pepper to taste and set yogurt sauce aside.
Place 1 rinsed can cannellini beans in large bowl and mash with potato masher. Add remaining rinsed can cannellini beans and mash until partially broken down. Whisk 1 large egg, remaining 1 tablespoon sun-dried tomato oil, and remaining ½ teaspoon salt together in separate bowl. Stir egg mixture, ⅓ cup panko bread crumbs, ⅓ cup finely chopped oil-packed sun-dried tomatoes, and remaining ¼ cup chopped basil into mashed beans until well combined. Divide mixture into 4 equal portions and lightly pack into ¾-inch-thick patties. Transfer patties to large plate and refrigerate for 10 minutes.
Add 2 tablespoons extra-virgin olive oil to hot baking sheet, tilting to coat. Arrange patties on half of sheet and add 3 ounces feta cheese, cut into ½-inch pieces, and 1 small red onion, halved and sliced ¼ inch thick to other half of sheet. Roast on lower rack until patties are browned on bottoms, about 8 minutes. Remove sheet from oven and heat broiler. Brush tops of burgers with remaining 1 tablespoon extra-virgin olive oil. Place sheet on upper rack and broil until patties are lightly browned on top, about 2 minutes.
Toss feta, onion, and 5 ounces baby spinach with 1½ tablespoons lemon juice in large bowl. Serve salad with patties and yogurt sauce.