Preheat the oven to 350 degrees F. Spray a 24-cup mini muffin tin with cooking spray.
Melt the butter in a small skillet over medium heat. Add the leeks and cook until lightly browned, 4 to 5 minutes. Set aside to cool, then divide them evenly among the muffin cups.
Whisk together the half-and-half, eggs and some hot sauce, salt and pepper in a medium bowl; mix in the ham. Fill the muffin cups two-thirds full with the egg mixture and top with the grated cheese. Bake until set, 8 to 10 minutes. Let cool for 5 minutes, then remove them from the tin.
To serve, stir together the sour cream and a couple dashes of hot sauce in a small bowl. Put a small dollop on each quiche along with a few thinly sliced chives.