Preheat oven to 350 degrees F (175 degrees C). Grease a 12-cup fluted tube baking pan (such as a Bundt(R) pan).
Combine white sugar and vegetable oil in the bowl of a stand mixer with a paddle attachment; beat in eggs and stir in vanilla extract.
Combine flour, baking soda, nutmeg, cinnamon, and salt in a bowl. Stir flour mixture into egg mixture until just blended. Stir in apples and pecans by hand to form a very heavy batter. Pour into the prepared pan.
Bake in the preheated oven until golden brown, about 1 hour and 10 minutes. Remove from the oven and leave cake in the pan.
Combine butter, brown sugar, milk, and 1 teaspoon vanilla extract in a small saucepan over medium heat. Bring to a boil and cook until a thickened glaze is formed, about 1 minute.
Pull the cake away from the sides of the pan with a fork. Pour glaze onto the cake immediately. Let cake cool in pan, about 10 minutes.