Preheat your oven to 350°F (175°C).
Bring a large pot of water to boil. Cut a small "X" on the bottom of each peach. Blanch peaches in boiling water for 30 seconds, then immediately transfer to an ice bath.
When cool enough to handle, peel the skin (it should slip off easily), remove pits, and slice into ½-inch wedges.
In a large bowl, gently toss the sliced peaches with granulated sugar, lemon juice, cornstarch, vanilla extract, cinnamon, and nutmeg until evenly coated.
In a medium bowl, combine oats, flour, brown sugar, salt, and cinnamon for the topping. Add the cold cubed butter and use your fingertips or a pastry cutter to work it into the dry ingredients until the mixture resembles coarse crumbs with some pea-sized butter pieces remaining.
Transfer the peach mixture to a 9x13 inch baking dish or a 10-inch cast-iron skillet, spreading it evenly.
Sprinkle the oat topping uniformly over the peaches, covering the entire surface.
Bake in the preheated oven for 40-45 minutes, until the filling is bubbling around the edges and the topping is golden brown.
Allow to cool for at least 5 minutes before serving.