Preheat oven to 350°F. Place baking dish in oven with a small pat of butter to heat.
In a large saucepan over medium heat, melt 4 tablespoons butter. Add peaches, ½ cup brown sugar, vanilla, salt, and ½ teaspoon cinnamon. Cook 6–10 minutes, until peaches soften and syrup thickens.
In a mixing bowl, whisk together sugar, self-rising flour, and milk, adding milk slowly to avoid lumps.
In another bowl, combine ½ cup brown sugar, all-purpose flour, ½ teaspoon cinnamon, and cold butter with a pastry cutter or fork until crumbly.
Remove hot baking dish from oven. Pour batter in first, then spoon peach mixture evenly over the top. Do not stir.
Sprinkle streusel topping evenly across the cobbler.
Bake 35–40 minutes, until golden brown and bubbling.
Serve warm with homemade ice cream or whipped cream.