Season a pot of water with enough salt so that it tastes like seawater and bring it to a boil. Add the potatoes and cook until forktender, about 12 minutes. Drain and transfer to a mixing bowl.
Stir in the following: capers, peppers, garlic, balsamic vinegar, sugar, red pepper flakes, and anchovies. (Use these even if you think you hate anchovies. They add a salty backbone, and any fishiness disappears into the dish.) Stir and let cool to room temperature.
Add the olive oil and 1/4 cup mint, then season with salt and pepper. Taste as you go-seasoning is personal. Sprinkle the Parmesan cheese and a few more torn mint leaves on top. Serve at room temperature. (It gets better after a day or two.)