Preheat your oven to 350°F (175°C). If using fresh peaches, peel and slice them. If using canned peaches, drain them well and set aside.
In a large bowl, combine the sliced peaches with 1/2 cup of granulated sugar, lemon juice, cinnamon, and vanilla extract. Toss to coat the peaches evenly and let them sit for about 10-15 minutes to draw out the juices.
In another bowl, whisk together the flour, remaining sugar, baking powder, and salt. Gradually stir in the milk and melted butter until the mixture is smooth and well combined.
Pour the batter into a greased 9x13-inch baking dish. Carefully spoon the peach mixture over the batter, making sure to pour the juices on top as well.
Bake in the preheated oven for 40-45 minutes, or until the cobbler is golden brown and the topping is set. A toothpick inserted in the center should come out clean.
Allow the cobbler to cool for a few minutes before serving. It's best served warm with a scoop of vanilla ice cream on top. Grandmas Peach Cobbler is a nostalgic and comforting dessert that many cherish.