Cook peaches, sugars, bourbon (optional), lemon juice, cinnamon and nutmeg over medium heat in a pot. Gently stir for about 10 minutes.
In a small bowl, stir the cornstarch and water. Add to peaches. Stir and cook 2 minutes. Take off of the heat and add vanilla. Allow to cool and store in fridge.
Cookies
In a pot with a stainless steel or white bottom, heat only 4 Tablespoons of the butter over medium heat. Swirl once in a while. Once you can smell the brown butter and brown bits form, take it off of the burner. Cool 30 minutes.
Add room temperature browned butter and the other 4 Tb. butter to a bowl. Add brown sugar. Beat together 1-2 minutes on medium. Add egg, vanilla. Beat until combined.
In large bowl, whisk together the flour, oats, cinnamon, baking soda and salt. Add to mixing bowl. Mix on low until combined.
Scoop in 1-2 Tb size portions onto a lined cookie sheet. Flatten the cookies slightly with greased spatula or bottom of a glass. Freeze until ready to bake.
Preheat oven to 350 degrees. Bake 8-10 minutes if room temperature. Bake 10-13 minutes if from frozen. They will be golden brown on the edges.
Frosting
With a fork, beat the yogurt (or butter) and cream cheese. Add the powdered sugar. Add salt and vanilla. Whisk with a fork until fluffy.
Assembly
Once the cookies are cooled to room temperature. Spread the frosting with an offset spatula. Using a slotted spoon, place a dollop of peaches in the middle. Serve room temperature. Store in the fridge 1-2 days.
The first day the cookies are crispy. The second day they get softer. We liked them both ways!