Preheat the oven to 350°F (175°C) and position the rack in the center. Lightly grease a 9x13-inch baking dish with butter or non-stick cooking spray.
Wash and peel the apples, then cut each in half and remove the core. Slice into 1/4-inch thick pieces and transfer to the prepared baking dish.
In a small bowl, whisk together granulated sugar, 2 tablespoons flour, cinnamon, and nutmeg. Drizzle lemon juice over the apples, then sprinkle with the sugar-spice mixture. Toss to coat evenly, spread in an even layer, and pour water over the top.
In a medium bowl, combine oats, 1 cup flour, brown sugar, cinnamon, and salt. Stir melted butter and vanilla together, then pour over dry ingredients. Mix with a fork until coarse, clumpy crumbs form.
Sprinkle the oat topping evenly over the apples, covering completely from edge to edge. Gently press some crumbs together to form clusters.
Bake on the center rack for 45 to 55 minutes, until the topping is golden brown and the apple filling is bubbling around the edges. Cover with foil if the topping browns too quickly.
Let the apple crisp cool for at least 15 minutes before serving. Serve warm in bowls with vanilla ice cream, whipped cream, or caramel sauce.