Preheat the oven to 375 degrees F. Add the butter directly to a 10x10 or similarly sized casserole dish and transfer to the preheated oven to melt. It will only take a few minutes to melt, so keep an eye on it.
Meanwhile, add sliced peaches, ½ cup white sugar and ¼ cup brown sugar to a large skillet. Heat over medium heat until the sugar is melted, stirring occasionally. This should take only 2-3 minutes. Once melted, turn off heat and squeeze in the fresh lemon juice.
In a medium bowl, mix together flour, ¾ cup white sugar, cinnamon, nutmeg and salt. Slowly pour in the milk, whisking until just combined. It should be about the consistency of muffin batter.
Add the batter to the melted butter, whisking gently to combine. Evenly pour the peach and sugar mixture onto the wet mixture. Don't stir.
Bake for 45-50 minutes, until the top is golden brown. Enjoy warm the same day, or keep for up to 3 days refrigerated. To refrigerate, first let the cobbler cool to room temperature, then store covered.
How to skin peaches: To easily remove the skin, simply add your peaches to a large pot of boiling water for 30 seconds. The peaches should be completely covered with water. Remove and immediately transfer to an ice water bath. After 30 seconds in the ice bath the skin should peel off easily.