Make the balsamic glaze: Bring the vinegar to a simmer in a saucepan over medium-high heat. Simmer until syrupy and reduced to about 1/2 cup, 8 to 10 minutes; keep warm or at room temperature.
Heat the oil in a heavy-bottomed pot or Dutch oven over medium heat until a deep-fryer thermometer registers 220 degrees F. Working in batches, add the potatoes and fry until slightly colored but still pale, about 5 minutes.
Transfer to a baking sheet using a large slotted spoon; let cool.
Increase the heat to bring the oil temperature to 350 degrees F. Working in batches, return the fries to the hot oil and fry until golden brown, 8 to 10 minutes.
Transfer to a paper towel-lined baking sheet using a large slotted spoon. Season immediately with salt and pepper.
Preheat the broiler.
Transfer the fries to a heatproof platter or baking sheet.
Sprinkle with the cheese and broil until it melts, about 2 minutes.
Drizzle with some of the balsamic glaze and sprinkle with the red pepper flakes. Photograph by Kat Teutsch
