Places potatoes in a pot of well-salted cold water and bring to a boil. Simmer for about 13 to 15 minutes (depending on size) until just tender. Drain well. When cool enough to handle, slice potatoes in half.

Heat 2 Tbsp. olive oil in a large skillet over medium high and arrange potatoes, cut side down, in a single layer, cooking them in batches if necessary. Let cook, undisturbed, until golden brown, about 3 to 4 minutes. Flip, and continue cooking another 2 minutes, until golden and crispy. Season liberally with kosher salt, and transfer cooked potatoes to a large serving platter.

To the same pan over medium-high heat, add 1 Tbsp. olive oil along with trimmed scallions. Weigh them down using a grill press or smaller pot to increase surface contact. Cook for 4 to 5 minutes, tossing halfway through, until nicely charred. Transfer to a cutting board.

Once cool enough to handle, finely chop the scallions and place in a medium mixing bowl. Add basil, remaining ¼ cup olive oil, vinegar, fish sauce, red pepper flakes, and ¼ tsp. salt. Mix to combine.

To the platter of potatoes, add sliced peaches and toss to combine. (Note: if you're adding arugula or micro-greens, gently incorporate them too.) Tear the burrata over the potatoes and peaches, letting the creamy interior drape overtop. Season burrata with a pinch of kosher salt. Spoon charred scallion dressing over all of the salad components, and garnish with extra torn basil leaves.