Preheat oven to 375°F (190°C). Lightly butter a 9-inch pie dish or 8x8 inch baking dish.

Prepare the filling: In a large bowl, gently toss the sliced peaches with the 3/4 cup sugar, 1/4 cup flour, lemon juice, vanilla extract, and cinnamon until the peaches are evenly coated. Pour the mixture into the prepared baking dish.

Prepare the topping: In a separate medium bowl, whisk together the 1 1/2 cups flour, 2 tablespoons sugar, baking powder, and salt.

Cut in the butter: Add the cubed cold butter to the dry ingredients. Use a pastry blender or your fingertips to cut the butter into the flour mixture until it resembles coarse crumbs, with some pea-sized pieces remaining.

Form the topping: Pour in the milk all at once and stir gently with a fork just until the dough comes together. Do not overmix. Drop spoonfuls of the dough evenly over the peach filling, leaving small gaps for the filling to bubble up.

Bake: Sprinkle the top generously with extra granulated sugar. Bake for 40 to 45 minutes, or until the topping is golden brown and the peach filling is thick and bubbly. If the edges begin to brown too quickly, cover loosely with foil.

Let cool for at least 20 minutes before serving warm, preferably with vanilla ice cream or fresh whipped cream.