Combine the potatoes, spinach, onion, curry powder, ginger, salt, and cayenne in a large bowl.
Heat olive oil in a large skillet over medium-high heat. Add the potato mixture and press into a layer. Cook, without stirring, until crispy and golden brown on the bottom, 3 to 5 minutes.
Reduce the heat to medium low. Fold in the chickpeas and broccoli, breaking up chunks of potato, until just combined. Press back into an even layer. Use a spoon to carve out 4 “wells” in the mixture. Break eggs, one at a time, into a cup or small bowl and slip one into each hole. Cover the skillet (if you do not have a lid that fits, a large sheet pan works well), and continue cooking until the eggs are set, about 4 to 5 minutes for soft, runny yolks.
Serve immediately with warm pita and Greek yogurt.