Pulse together the Swiss chard, chickpeas, garlic, cumin, ½ teaspoon salt, and ¼ teaspoon pepper in a food processor until finely chopped, scraping down the sides of the bowl as necessary.
Combine the Swiss chard mixture, Feta, and flour in a large bowl. Form the mixture into eight 2½-inch patties.
Heat 2 tablespoons of the oil in a large nonstick skillet over medium-high heat.
Cook the patties, in 2 batches, until browned, 3 to 4 minutes per side, adding the remaining 2 tablespoons of oil to the skillet for the second batch. Serve with the yogurt and hot sauce.