Preheat the oven to 350°F (175°C). Lightly grease a 9x9-inch baking dish.
In a large mixing bowl, stir peeled, sliced peaches with 1/4 cup lemon juice, 1/4 cup granulated sugar, 2 tablespoons cornstarch, 1/2 teaspoon ground cinnamon, and 1/2 teaspoon vanilla extract until well combined. Transfer this mixture into the prepared baking pan, including all the juices.
In a medium bowl (or using a food processor), mix together 1 1/2 cups all-purpose flour, 3/4 cup brown sugar, and 1/2 teaspoon ground cinnamon. Add the cold, cubed butter and use a pastry cutter (or pulse in the food processor) to combine the mixture. The mixture should be crumbly, with small clumps of butter remaining.
Evenly sprinkle the crumble topping over the peaches in the baking dish, covering them completely.
Bake for 40-45 minutes, or until the crumble topping is golden brown and the peach filling is bubbling. Serve warm.
Notes
To peel peaches easily, blanch them in boiling water for just a few seconds, then immediately plunge them into a bowl of ice water. The peels will naturally loosen.
For frozen peaches, thaw at room temperature for about 30 minutes before using. Double the amount of cornstarch in the recipe to compensate for the extra moisture.
For canned peaches, choose ones canned in light syrup or no syrup for best results. Rinse off any syrup in cold water. Double the amount of cornstarch in the recipe to compensate for the extra moisture.