Remove the goat cheese from its package, place in a bowl and let sit at room temperature to soften. Cut the peaches in half along the seam that runs around the fruit. Remove the stone. Drizzle the halves lightly with the oil and season with salt and pepper.
Heat the grill. Place a few peaches on the grill, cut-side down, and grill for just under 1 minute. As you grill, press down firmly with tongs for a couple of seconds to form grill marks. Turn and cook for 10 seconds on the second side. Repeat with the remaining peaches.
Add a dash of water to the goat cheese and stir until pliable. Using two spoons, place a generous spoonful of goat cheese in the indentation of each peach, scooping the cheese with the first one and using the second to push the cheese onto the fruit. Spoon a few drops of the balsamic vinegar over the peaches and season with more pepper. Serve immediately.
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