Peel and dice the fresh peach into ½-inch cubes, removing the pit. Set aside about ½ cup of diced peach.
In a 12-14 oz microwave-safe mug, add the melted butter, milk, vanilla extract, and granulated sugar. Stir with a fork until the sugar begins to dissolve.
Add the flour, baking powder, and salt to the mug. Stir gently with the fork until just combined and no dry flour remains — don't overmix or the cake will be tough.
Gently fold in the diced peaches, distributing them throughout the batter. Some will sink to the bottom, which creates the classic cobbler effect.
Microwave on high power for 90 seconds to 2 minutes, checking at 90 seconds. The top should look set and spring back when lightly touched, and the edges will pull slightly away from the mug.
Let the mug cake rest for 30-60 seconds to allow the steam to finish cooking the center and the peach juices to thicken.
Serve warm directly from the mug with a spoon. Top with vanilla ice cream, whipped cream, or enjoy as-is while the peaches are still bubbling.
