Preheat oven to 375 degrees F (190 degrees C). Pour 2 teaspoons melted butter into the bottoms of two 6-ounce glass or ceramic ramekins.

Combine self-rising flour with 1/2 cup sugar in a bowl; whisk in milk to make a smooth batter. Whisk Parmigiano-Reggiano cheese and black pepper into the batter; divide equally between the prepared ramekins.

Place peach slices into a bowl and top with 2 tablespoons sugar and basil. Drizzle with balsamic vinegar and water; mix. Allow peaches to rest and let sugar draw out the moisture, about 10 minutes. Divide sliced peaches and their juice over the batter.

Bake in the preheated oven until the cobbler batter rises up over the peaches and cobblers are browned and bubbling, about 35 minutes. Let cool for about 20 minutes before serving for cobblers set up. Serve warm.