Add the feta cheese, cream cheese, Greek yogurt, olive oil, and water to a small food processor. Blend until thick, creamy, and spreadable, with small flecks of feta remaining.
Place the ground chicken, panko breadcrumbs, hot sauce, garlic powder, and salt in a medium bowl. Gently mix until evenly combined without overworking the chicken.
Divide the mixture into 4–5 portions and shape each portion into a burger patty.
Heat a small amount of oil in a large nonstick skillet over medium-high heat. Add the patties, leaving enough space between them to cook evenly.
Cook for approximately 3–4 minutes per side, or until golden on the outside and the centers reach a safe internal temperature of 165°F (74°C).
Stir the melted butter and remaining hot sauce together. As the Buffalo Chicken Burgers finish cooking, generously brush both sides with the buttery hot sauce mixture.
Spread whipped feta over the burger buns. Add lettuce and a cooked chicken patty, then top with the remaining half of the bun.
Serve immediately while the burgers are hot and the whipped feta is cool and creamy.