Preheat your oven to 350°F.
Take your half cup of butter and place it into a stainless or light bottomed saucepan/skillet
Brown the butter: Allow the butter to melt. The butter will then begin to foam and then subside, and then bubble. After the butter is done foaming, stir the butter with a rubber spatula or swirl it around the pan. When the milk solids begin to turn a golden brown and the butter begins to have a nutty fragrance, remove from heat and pour into a heat proof bowl, making sure to scrape the toasted milk fats into the butter as well. Place in the refrigerator to cool.
Take your other 1 tbsp of butter and grease your cast iron skillet or pan with the butter.
Cut your peaches, removing the pits and either slice or dice. You can remove the skin if you prefer but it is not necessary.
Place all of your cut peaches into a bowl and add cornstarch, granulated sugar and vanilla. Toss to coat the peaches.
Add your peaches to the greased skillet or pan.
In a medium sized bowl, add your oats, flour, cinnamon, brown sugar, and salt. Stir to mix.
Pour in your browned butter and stir, making sure all the filling gets coated with the butter.
Cover the top of your peaches with the crisp topping in an even layer.
Place your skillet or pan in the oven on the center rack and bake for 30-40 minutes. If your topping starts to get too dark, cover with tinfoil for the remaining bake time.
Remove from oven and let cool for at least 30 minutes before serving. The resting time will allow the juices from the peaches to become less runny.
Serve with vanilla ice cream, and store any leftovers in the refrigerator covered for 1-2 days. Enjoy!