Preheat the oven to 350°F (175°C) and lightly grease an 8-inch square baking dish or a similar-sized oven-safe dish.
Drain the canned peaches well and place them in a medium bowl.
Add the sugar, cornstarch, vanilla extract, cinnamon, and optional lemon juice and nutmeg. Stir gently until the peaches are evenly coated.
In a separate bowl, whisk together the flour, sugar, baking powder, salt, and cinnamon.
Add the milk, melted butter, and vanilla extract. Stir just until combined to form a smooth, thick batter. Avoid overmixing.
Spread the peach filling evenly across the bottom of the prepared baking dish.
Spoon the topping over the peaches and spread it gently. Leave a few peaches visible if desired.
Bake for 35–45 minutes, or until the topping is golden brown and cooked through and the peach filling bubbles around the edges.
Remove the cobbler from the oven and let it rest for 10–15 minutes before serving.
Serve warm with vanilla ice cream or whipped cream, if desired.