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

Sift flour, white sugar, salt, and baking soda together into a large bowl; make a well in the center. Stir oil, beaten eggs, and vanilla extract into the well until well combined (batter will be thick); fold in apples and walnuts until evenly distributed. Spread batter into the prepared baking dish.

Bake in the preheated oven until a toothpick inserted into center comes out clean, 40 to 50 minutes.

Combine brown sugar, butter, and milk in a small saucepan; bring to a boil. Cook, stirring constantly, until glaze is slightly thickened, 2 to 3 minutes.

Pour glaze over hot cake.