Preheat the oven to 375°F. Lightly grease two 6- to 8-ounce baking dishes or ramekins.
Combine the chopped peaches, brown sugar, vanilla or almond extract, and cornstarch in a small bowl. Stir until the peaches are evenly coated and the cornstarch is no longer visible.
Divide the peach mixture evenly between the prepared baking dishes.
In another small bowl, combine the oats, flour, cinnamon, and salt. Add the cold butter and mix with a fork or your fingertips until the mixture forms small crumbs and clusters.
Sprinkle the oat topping evenly over the peach filling.
Place the baking dishes on a lined, rimmed baking sheet to catch any bubbling juices.
Bake for 15 to 20 minutes, or until the topping is golden brown and the peach filling is bubbling around the edges.
Let the crisps cool for 5 to 10 minutes before serving. Serve warm, preferably with vanilla ice cream.