Preheat the oven: Preheat to 350°F and grease a 9×9-inch baking dish.
Make the oat topping: In a large bowl, stir together oats, brown sugar, flour, cinnamon, and nutmeg. Add melted butter and mix until crumbly. Press half of the mixture into the bottom of the prepared baking dish.
Layer the fruit: Evenly distribute the sliced peaches and strawberries over the oat layer.
Make the syrup: In a small saucepan, whisk together brown sugar, granulated sugar, cornstarch, salt, and water. Cook over medium heat, stirring constantly, until thickened. Remove from heat and stir in vanilla extract.
Add the syrup: Pour the warm syrup evenly over the fruit layer.
Top and bake: Sprinkle the remaining oat mixture over the top. Bake uncovered at 350°F for 50 minutes, until bubbly and golden brown.
Optional: Broil for 2–3 minutes at the end for an extra crispy top—watch closely to avoid burning.
Cool and serve: Let cool for 10–15 minutes before serving. Enjoy warm with vanilla ice cream.