Preheat the oven to 375°F (190°C). Grease or line a baking dish with parchment paper.
In a large mixing bowl, toss the peaches with sugar, vanilla, cinnamon, cornstarch, and lemon juice, ensuring the slices are evenly coated. Transfer the mixture to the prepared dish, spreading it evenly.
For the topping, combine flour, oats, brown sugar, granulated sugar, baking powder, and salt in a medium bowl. Pour in melted butter and stir until the mixture forms a crumbly texture.
Distribute the crumble topping evenly over the peach mixture. For a savory twist, sprinkle shredded cheddar cheese on top if desired.
Bake for 40–45 minutes or until the topping turns golden brown and the filling is bubbling.
Allow the cobbler to cool slightly before serving. Enjoy warm, paired with a scoop of vanilla ice cream or a dollop of whipped cream.