Preheat the oven to 350 degrees F (175 degrees C). Grease and flour a 9x13-inch baking pan.

Place oats into a bowl and cover with boiling water. Set aside.

Sift flour, pudding mix, cinnamon, salt, baking soda, cloves, and allspice together in a small mixing bowl. Set aside.

Beat brown sugar, white sugar, and butter together using an electric mixer until fluffy. Beat in eggs, 1 at a time, until combined. Stir in flour mixture and coconut until blended; stir in softened oats and apples until well blended, making sure not to overmix. Pour batter into the prepared pan.

Bake in the preheated oven until lightly browned on top and a toothpick or knife inserted into the center comes out clean, 35 to 40 minutes.

While cake is baking, melt butter in a medium saucepan over medium-high heat. Boil, stirring constantly, until butter begins to brown, 2 to 3 minutes. Reduce heat to low and stir in brown sugar and coconut milk until well blended; stir in coconut and pecans. Remove from heat.

Remove cake from the oven and immediately spread butter mixture over the top. Set an oven rack 2 to 4 inches from the heat source and preheat the oven's broiler.

Broil cake in the preheated oven until topping bubbles and browns, about 2 minutes, making sure not to burn. Remove pan from the oven. Let topping absorb into the cake before serving, about 10 minutes.