In a medium bowl, whisk the flour with the oats, the 3 sugars, the salt, cinnamon and nutmeg.
In a small saucepan, melt the butter over moderately low heat, stirring, until it turns a deep golden color and it will also smell gorgeous and nutty. This should only take about 8 minutes -- but watch it! You don't want burnt butter.
Scrape the browned butter into the flour mixture, and don't forget to include all the tasty browned bits at the bottom of the pan!
Stir mixture until combined.
Scatter the topping evenly over the peaches.
Bake for 50 to 60 minutes, until the crisp is golden and bubbling.
Transfer to a rack and let stand for 15 minutes before serving with vanilla ice cream.
Why is the waiting time important? Because you want the cobbler to cool, so everything doesn't run all over your plate. It's the same reason you let cookies or brownies sit for a few minutes when they're fresh from the oven.