Preheat the oven to 375°F and generously butter a medium sized baking dish (we used an 11 x 7 oval, but a 9 x 9 or a 10 inch skillet would also be great!).
Pit and thinly slice the peaches into half-moon shapes about ¼ inch thick. Place the peaches in your prepared dish and top with brown sugar, flour and vanilla extract. Use your hands or a wooden spoon to toss to combine. Once the peaches are evenly coated, gently spread them back into an even layer. Set aside.
Next, make the crisp topping. Combine the flour, oats, brown sugar, granulated sugar, salt, and butter in a medium bowl and use your fingers or a fork to stir everything together.
Crumble the crisp topping over the peaches to evenly coat the top.
Bake for 40 to 45 minutes or until the peaches are bubbling and the crisp is golden brown.
Allow to cool slightly before serving warm! Top with vanilla ice cream or whipped cream for the ultimate treat!