Preheat oven to 175°C (350°F). Lightly grease a 9x13-inch baking dish.

Peel, core, and slice apples into 5mm wedges. Place in a large bowl with brown sugar, granulated sugar, flour, cinnamon, nutmeg, salt, lemon juice, and vanilla. Toss until each apple slice is evenly coated. Transfer to the prepared baking dish.

In a separate bowl, combine rolled oats, flour, brown sugar, salt, and cinnamon. Stir briefly to distribute. Add cold cubed butter and work it into the dry mixture by pressing and crumbling with a pastry cutter or clean fingertips until irregular clumps form and no large pieces of unmixed flour remain. Stop before the mixture becomes uniform.

Scatter the topping loosely over the apple filling in even handfuls, covering edge to edge without pressing down.

Bake on the center rack for 40 to 45 minutes until the topping is deep golden-brown and the filling is actively bubbling at the sides of the dish. If the topping browns early, tent with foil for the remaining bake time, then uncover for the last 5 minutes.

Transfer to a wire rack and rest for 10 minutes before serving. Serve warm with vanilla ice cream or whipped cream.