Preheat the oven to 375°F. Place an ovenproof 9x13-inch baking dish on the counter. In a small saucepan over low heat, melt the stick of butter until just liquefied, then swirl or pour it into the baking dish so the bottom is evenly coated; set the dish aside while you make the batter.
Make the batter: in a medium bowl whisk together the flour, 3/4 cup brown sugar, baking powder, cinnamon, and salt until uniform. In a separate bowl combine the milk and vanilla. Pour the wet into the dry and stir gently with a rubber spatula until just combined—the batter should be slightly lumpy, do not overmix.
Prepare the peaches: toss the peach wedges in a bowl with 1/3 cup brown sugar, 2 tablespoons maple syrup and the lemon zest. Let them macerate 5 minutes so juices begin to draw out.
Arrange half of the peach wedges in an even layer over the melted butter in the baking dish. Spoon the batter by large spoonfuls over the peaches and gently spread to cover as evenly as possible (the batter will flow around the fruit). Scatter the remaining peaches across the top; don’t press them in—this creates the classic cobbler texture.
Let the assembled dish sit for 8–10 minutes at room temperature—this helps the batter settle around the fruit and creates a better crust during baking.
Bake on the center rack for 42–48 minutes, until the top is golden and the center is just set (a toothpick inserted into batter should come out mostly clean with a few moist crumbs). If the edges brown too quickly, tent loosely with foil for the last 10 minutes.
While the cobbler bakes, make the maple-butter drizzle: combine 3 tablespoons butter and 1/4 cup maple syrup in a small saucepan. Bring to a gentle boil and cook for about 90 seconds, stirring, then remove from heat—this slightly thickens the sauce and concentrates flavor.
Remove the cobbler from the oven and let it rest 10 minutes so juices thicken slightly. Serve warm or at room temperature with vanilla ice cream and a generous drizzle of the maple-butter sauce.