Preheat your oven to 375°F (190°C). Lightly butter or spray a 9x9 inch baking dish or an equivalent 2-quart baking dish.

In a large bowl, combine the sliced peaches, 0.5 cup granulated sugar, cornstarch, lemon juice, vanilla extract, ground cinnamon, and 0.25 teaspoon salt. Toss gently until the peaches are evenly coated. Pour the peach mixture into the prepared baking dish and spread it into an even layer.

In a separate medium bowl, whisk together the all-purpose flour, rolled oats, brown sugar, 0.25 cup granulated sugar, and 0.25 teaspoon salt.

Add the cold, cubed butter to the dry topping ingredients. Use a pastry blender, two forks, or your fingertips to cut the butter into the flour mixture until it resembles coarse crumbs with some pea-sized pieces of butter remaining. Don't overmix; you want some texture here!

Evenly sprinkle the crisp topping over the peach filling in the baking dish.

Bake for 35-45 minutes, or until the topping is golden brown and the peach filling is bubbly and tender when poked with a knife. If the topping starts to brown too quickly, you can lightly tent it with aluminum foil.

Remove the crisp from the oven and let it cool on a wire rack for at least 15-20 minutes before serving. This allows the filling to set up nicely. Serve warm, perhaps with a scoop of vanilla ice cream or a dollop of whipped cream!