1. Microwave potatoes on high until tender (about 15 minutes).
2. Meanwhile, in a medium saucepan, combine vinegar, sugar, and peppercorns; bring to a boil and stir to dissolve sugar. Place onion slices in a medium heatproof bowl; pour hot vinegar mixture over onion and let stand 10 minutes. Drain onion slices and transfer to a plate to cool.
3. Split cooked potatoes open with a knife; remove flesh and transfer to a bowl (you should have about 1 cup). Use a fork to mash potato with salt and pepper; divide potato filling among 4 tortillas. Top each with 1 heaping cup spinach, 1/4 cup cheese, and another tortilla.
4. Heat a large nonstick skillet over medium-high heat. Working in batches, heat 1 teaspoon oil and transfer 1 quesadilla to skillet; cook until quesadilla is crisp and cheese is melted (about 3 minutes per side). Transfer cooked quesadilla to serving plate. Repeat with remaining oil and quesadillas. Cut quesadillas in quarters and serve each with 1/4 cup pickled onions.