In a large bowl, beat the sugar, oil, eggs and vanilla until well blended. In a large bowl, combine the four, baking soda, salt and nutmeg; gradually add to the egg mixture until well blended(batter will be stiff). Stir in apples and walnuts.
Spread into a greased 13x9-in. baking pan. Bake at 350° for 45-50 minutes or until a toothpick inserted in the center comes out clean. Cool on a wire rack.
For the sauce, in a small saucepan, combine the milk, sugar, flour and butter. Bring to a boil over medium heat; cook and stir for 2 minutes. Remove from the heat; stir in vanilla. Cut cake into squares; serve with warm sauce. Garnish with walnut halves if desired.