Preparation

Make the ice cream

• Peel and pit the peaches. Cut into ½-inch pieces. In a medium saucepan, combine the peaches and sugar. Cook over medium heat, stirring frequently, until the fruit becomes juicy and the sugar dissolves, about 5 minutes. Add more sugar to taste if the peaches seem tart. Continue to cook the peaches, stirring frequently, until thickened and jammy, 10 to 20 minutes. (If the peaches are hard, you may have to add water during cooking to help them break down. If they are soft and release a lot of juice, you will need to cook them longer.) Put the peach jam in the bowl of a food processor, then purée until smooth. Refrigerate until cold.
• Put an 8×8- or 9×9-inch pan in the freezer. If the ice cream was frozen after churning (or if you’re using store-bought ice cream), let it sit at room temperature for about 10 minutes to soften, then put it in the bowl of a stand mixer fitted with the paddle attachment. With the mixer on medium-low speed, stir in 1/2 to 2/3 cup of the jam, just until combined (Alternatively, fold together in a bowl using a rubber spatula.) Spread the ice cream into the frozen pan, cover with plastic wrap directly on the surface of the ice cream, and freeze until scoopable, about 4 hours or overnight.

Make the muffins

• Position a rack in the center of the oven, and heat the oven to 375°F. Lightly butter the cups of a 12-cup muffin tin.
• In a large bowl, whisk the milk, sour cream, eggs, and vanilla. Set aside.
• In another large bowl, whisk the flour, granulated sugar, baking powder, and salt.
• Using your fingers, mix the cold butter into the dry ingredients until the butter is the size of small peas. Add the milk mixture, and stir gently until almost smooth. Gently fold in the blueberries.
• Spoon the batter into the prepared tins. Sprinkle the turbinado sugar over the batter. Bake until a skewer inserted into the middle of a muffin comes out clean, about 15 minutes. Cool for 5 minutes, then gently remove from the tin. Cool completely.

Assemble the sandwiches

• Cut the muffins in half horizontally. Put the muffin bottoms on a platter or plate. Put a scoop of ice cream on top, and then place the top half of the muffin on the ice cream. Gently press down on the muffin top to adhere the sandwiches together. Serve immediately.