For the peach cobbler filling: In a large pot, bring the peaches along with their syrup to a boil over medium heat. Add the sugar, butter, vanilla, nutmeg and cinnamon and mix well. Reduce the heat and let simmer on low so the flavors can penetrate the peaches, about 15 minutes.
Meanwhile, in a separate bowl mix the flour with 1/4 cup warm water. Slowly add to the simmering peach and spice mixture and stir until it thickens to the consistency of gravy, 2 to 3 minutes. Turn off the heat.
Lightly coat a 13-by-9-inch baking pan with nonstick cooking spray.
For the crust: On a lightly-floured surface, roll out one pie crust to fit the bottom and up the sides of your baking pan. Ease the crust into the baking pan, then pour the cobbler mixture on top of the crust, being careful not to fill the baking dish to the very top. Roll out the other pie crust and slice into strips. Arrange the strips of dough across the cobbler both lengthwise and widthwise to create a weaved look.
Bake until the crust is golden brown, 40 to 45 minutes.