Place burrata on a layer of paper towels and let it rest at room temperature for about 20 minutes to enhance its creaminess.
While the cheese warms, halve or coarsely crush cherries: press each cherry lightly under the flat side of a knife or bottom of a glass until it splits, then remove the pit and reserve the juices.
Tear the burrata into large chunks and arrange on a serving platter. Scatter the prepared cherries and any collected juices over the cheese.
Drizzle generously with extra-virgin olive oil. In a dry skillet over medium heat, toast the pistachios for 2–3 minutes, stirring, until fragrant and slightly golden; let cool, then roughly chop.
Sprinkle the toasted pistachios over the burrata and cherries. Finish with flaky sea salt, freshly ground black pepper, and tear the mint leaves over the top.
Serve immediately with warm crostini and watch your guests dive in.