Peel the peaches and slice them into 3/4-inch thick pieces. Transfer the slices to a medium-sized bowl, sprinkle with granulated sugar, and toss to coat. Let the peaches sit for around 20 minutes to allow the sugar to extract some of their juices.
In another bowl, combine the flour, brown sugar, oats, baking powder, cinnamon, nutmeg, and salt for the topping. Using a pastry blender or fork, cut the cold butter into the mixture until it forms small crumbs. Place the topping mixture in the refrigerator to chill.
Preheat the oven to 375°F (190°C).
Drain the peaches, reserving the juice. Transfer the peaches to a large mixing bowl. Measure 1/4 cup of the peach juice and mix it with the flour, lemon juice, vanilla extract, salt, and cinnamon in a small bowl. Pour this mixture over the peaches, stirring to combine.
Place the coated peach slices into an 8x8-inch or similarly sized baking dish. Evenly distribute the oat crumble over the peaches.
Bake the crisp for 30-35 minutes or until the topping is golden brown and set. Allow the dessert to cool for at least 10 minutes before serving.
Serve warm, accompanied by vanilla ice cream if desired.