Preheat the oven to 350°F.
Place the butter in a 2 ½- to 3-quart casserole dish. Put the dish in the oven for several minutes, just until the butter has completely melted. Carefully remove the dish from the oven and set it aside.
Add the sliced peaches, granulated sugar, lemon juice, and cinnamon to a large skillet.
Cook over medium heat for 2 to 3 minutes, stirring gently, just until the sugar has melted and the peaches begin releasing their juices. Remove the skillet from the heat.
In a medium mixing bowl, whisk together the self-rising flour and granulated sugar.
Add the milk and whisk just until the batter is smooth and no dry flour remains. Do not overmix.
Pour the batter evenly over the melted butter in the casserole dish. Gently spread it with a silicone spatula, but do not stir the batter into the butter.
Spoon the peaches evenly over the batter, then drizzle all of the syrup from the skillet over the top. Do not stir the layers together.
Sprinkle the coarse sugar over the cobbler, if using.
Bake for 40 to 45 minutes, or until the top is deeply golden brown, the fruit is bubbling around the edges, and the center appears fully baked.
Let the peach cobbler cool for at least 15 minutes before serving. Serve warm on its own or with vanilla ice cream.