Place sliced onion in a jar or other heatproof lidded container. In a small saucepan, combine vinegar, 3⁄4 cup water, sugar, and salt. Bring to a simmer and stir. Once sugar and salt have dissolved, pour the hot brine directly over the onions. Cover and set aside to cool.
Coat the bottom of a large skillet with oil over medium heat. Once hot, add halloumi and cook until deeply browned and crisp on the bottom, about 3 to 4 minutes. (When first added to the skillet, they will spatter loudly—they’re browning once spattering quiets to a sizzle.)
Flip halloumi strips and brown for 1 to 2 more minutes. Set aside to cool.
Turn heat up to medium-high, adding another drizzle of oil if pan looks dry. Cook peaches in a single layer, cut sides down, until deeply caramelized and charred—don’t be scared if they look almost burnt. Flip and continue to char all cut sides. Remove from heat and set aside to cool. Reserve any oil or juices that accumulated in the skillet.
While peaches and halloumi cool, add corn and mint to a large bowl.
Once cool to slightly warm, cut halloumi and peaches into 1⁄2-inch pieces and add to the bowl, along with arugula, 3⁄4 cup pickled onions, 2 tablespoons pickling liquid, a generous pinch of salt, a drizzle of olive oil, and reserved skillet oil and juices.
Toss to combine, adding more salt or pickling liquid to taste. Serve immediately.