Rinse potatoes and place them in a large pot.
To the pot, add vinegar, then add enough water to cover the potatoes by one inch. Add ½ teaspoon salt. Bring to a boil and cook until potatoes are very tender, but not falling apart, about 25-30 minutes, depending on the size of the potatoes. Drain vinegar water off.
Heat oven to 450 F.
Return potatoes to pot and add butter. Gently stir to melt butter and coat potatoes.
Brush large sheet pan with olive oil. Spread out cooked potatoes on sheet pan.
Using the bottom of a flat measuring cup or mug, smash potatoes to ½-inch thickness.
Roast in oven until golden brown on bottom, about 15-20 minutes, then flip potatoes and roast for another 15-20 minutes.
Remove from oven and drizzle with balsamic reduction. Sprinkle with chopped basil and coarse sea salt to taste. Serve warm or cold.