Preheat oven to 325 degrees F (165 degrees C). Grease a fluted tube pan (such as Bundt(R)).
Beat yogurt, white sugar, 1 cup brown sugar, eggs, and vanilla extract together in a bowl using an electric mixer until smooth.
Combine flour, flax seed, cinnamon, salt, cloves, and nutmeg together in a separate bowl. Stir yogurt mixture into flour mixture just until batter is combined; fold in apples. Pour batter into prepared pan.
Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, 70 to 75 minutes. Turn cake onto a serving plate.
Melt butter, 1/2 cup brown sugar, and milk together in a saucepan over medium heat until sugar is dissolved, 2 to 3 minutes. Remove saucepan from heat and cool until sauce is thickened, about 10 minutes. Pour sauce over cake.