Heat a large cast-iron skillet over medium-high heat for 20 minutes.
Meanwhile, gently toss together the chuck and sirloin in a large bowl, making sure not to overwork the mixture.
Combine the paprika, coriander, cumin, garlic, mint, brown sugar and cinnamon in a small bowl. Sprinkle the spice mixture over the beef; add the crumbled feta and fold gently to combine. Form the beef into four 3/4-inch-thick patties, each about 4 inches in diameter. Sprinkle both sides generously with salt and a few grinds of pepper.
Place the patties in the hot skillet; do not press down with a spatula. When the patties are seared and browned on one side, flip and cook on the other side until browned, about 3 1/2 minutes per side for medium-rare.
Assemble the burgers using your favorite buns, toppings and condiments.