Preheat your oven to 350°F (175°C). Place the stick of butter in a 9x13-inch baking dish and let it melt in the oven while preheating. Keep an eye on it to ensure it doesn’t burn.
In a medium bowl, whisk together the flour, baking powder, salt, and sugar. Add the milk and vanilla extract, stirring until smooth. Set aside.
In a small saucepan, combine the lemon juice, lemon zest, sugar, and water. Heat over medium heat, stirring until the sugar dissolves. Remove from heat and let it cool slightly.
Once the butter is fully melted, carefully remove the baking dish from the oven. Pour the batter evenly over the melted butter—do not stir! Next, gently pour the lemon filling over the batter. Again, do not stir. The batter will rise around the lemon filling as it bakes.
Place the baking dish in the oven and bake for 40–45 minutes, or until the top is golden brown and set. The center may still be slightly jiggly, but it will firm up as it cools.
Allow the cobbler to cool for at least 15–20 minutes. Serve warm with a scoop of vanilla ice cream or a dollop of whipped cream for an extra indulgent treat.