Arrange a rack in center of oven; preheat to 425°. In a medium bowl, whisk syrup, miso, vinegar, and soy sauce until smooth. Add water and stir to combine.
In a large bowl, toss potatoes, 1 Tbsp. oil, and ½ tsp. kosher salt until coated.
In a large (preferably cast-iron) ovenproof skillet over medium-high heat, heat remaining 1 Tbsp. oil. Add potatoes cut sides down and cook, undisturbed, until golden brown on the bottom, 2 to 3 minutes. Flip potatoes; season second side with remaining 1 tsp. kosher salt. Cook, undisturbed, until golden on the other side, 1 to 2 minutes more.
Add butter to skillet and let melt. Using an oven mitt or towel, tilt pan slightly so butter pools near the bottom. Using a large spoon, spoon frothing butter over potatoes until shimmering. Add maple-miso glaze to pan (it should reach about halfway up potatoes) and bring to a simmer.
Transfer skillet to oven and roast until liquid is reduced and potatoes are tender, 23 to 28 minutes.
Transfer potatoes to a platter. Whisk sauce in pan to incorporate and drizzle over potatoes. Sprinkle with sea salt (if using).
