Whisk the flour, baking powder, and salt together in a large mixing bowl.
Add the fat and cut it into the flour with a pastry cutter or fork until the mixture looks like coarse, sandy crumbs.
Add the water and sourdough starter and mix until a shaggy dough forms, then knead for 2 to 3 minutes until smooth. If the dough feels crumbly, add water 10 grams at a time, and if it feels sticky, add flour 10 grams at a time.
Cover the bowl tightly and let the dough ferment at room temperature until noticeably puffed and roughly doubled, 6 to 8 hours for a same day ferment or 8 to 10 hours overnight.
Divide the dough into 10 equal pieces, about 69 grams each. Preshape each piece into a tight, smooth ball, cover, and let the balls rest for 10 to 15 minutes.
Preheat the waffle iron to a medium setting and butter both plates well.
Roll one dough ball on a lightly floured surface until it is just slightly smaller than your waffle iron, about 7 inches across for a 7 1/2 inch iron.
Whisk one or two eggs in a small bowl with a pinch of salt and pepper, then pour it onto the center of the bottom plate so it spreads into the grid.
Lay the rolled tortilla directly on top of the egg and close the lid right away.
Cook for 3 to 4 minutes, until the tortilla is golden and the egg is fully set.
Lift the egg tortilla out and lay it egg side up. Fill and roll it while it is still warm so it rolls without cracking.
Butter the plates again and repeat with the remaining dough balls and eggs, rolling the next tortilla while the previous one cooks.