In a large bowl, stir together flour, salt, and 1⁄2 cup water until a dough forms. Transfer dough to a lightly floured work surface and knead until smooth, 4-6 minutes. Divide dough into 6 pieces, roll into balls, and transfer to a round 9" dish. Pour 1 cup ghee over dough balls and cover dish with plastic wrap; let rest for 2 hours.
Meanwhile, heat 2 tbsp. ghee in a 4-qt. saucepan over medium heat. Add mint, garam masala, coriander, chile flakes, cumin, garlic, and ginger and cook until fragrant, 1-2 minutes. Add beef, tomato, and yogurt and cook, stirring, for 10 minutes. Remove from heat and stir in cilantro, chiles, red onion, salt, and pepper; set meat mixture aside to let cool.
Transfer dough balls to a work surface, reserving ghee in the baking dish. Working with one ball at a time, roll dough into a 10" circle. Brush 1 tsp. of the reserved ghee onto dough and dust with 1 tsp. flour. Starting from edge closest to you, begin making 1⁄2" accordion-like folds, laying one fold on top of the other until you have a single long, stacked strip of dough. Leaving dough strip on its side, so that the accordion pleats face you, twist the strip of dough into a spiral, tucking the loose end underneath. Using a rolling pin, flatten the spiral into an 8"-10" circle. Place one-sixth of the meat mixture in the center of the dough and form a well. Crack one egg into the center of the meat mixture. Fold edges of the dough over the egg so that the edges meet in the middle; pinch edges together to form a closed square packet. Transfer dough packet to a baking sheet, cover with a moist towel, and set aside; repeat with remaining dough circles, ghee, flour, meat mixture, and eggs; set dough packets aside.
Working in 3 batches and wiping out skillet between each batch, heat 2 tbsp. of remaining reserved ghee in a 12" nonstick skillet over medium heat; add two of the dough packets and cook, flipping once, until golden brown and the egg inside has just set, about 12 minutes. Repeat with remaining ghee and dough packets.