In the bowl of a stand mixer, combine yeast, warm water and about a teaspoon of sugar. Let this sit for about 5 minutes, or until foamy.
Add the milk, remaining sugar, salt, melted butter and egg to the bowl. Stir everything with a wooden spoon until incorporated.
Attach the dough hook to the stand mixer. Add half of the flour and beat until the flour is moistened. Add the remaining half and beat until all the flour is incorporated. The mixture is now a soft, wet and sticky dough. Beat the mixture for 5-7 minutes, sprinkling up to about 4-5 tbsp. of flour, until the dough is smooth and has gathered in the center of the bowl. It should be slapping the sides of the bowl as the mixer runs.
Dust a bowl lightly with flour, as well as your hands. Shape the dough into a smooth ball with your hands and place it in the dusted bowl. Cover and let this rise for about 1½ hours, or until it has doubled in bulk.
On a lightly dusted surface, use a rolling pin to roll the dough into a ¼ inch thick rectangle. Trim the edges to make the rectangle straight. Spread the mayonnaise all over the dough. Arrange the ham slices across the surface then top with the cheese slices. Make sure the entire surface is covered with the ham and cheese.
With the long edge nearest you, roll the dough into a tight log. Cut into 16 portions and place on a parchment-lined 9 x 13 inch baking pan. Cover the pan loosely and the the buns rise for 40 minutes to 1 hour.
Preheat oven to 350 F. Bake the rolls at 350F for for 25-30 minutes or until lightly golden. In a small combine melted butter garlic powder. Brush the glaze all over the buns, then sprinkle parsley. Serve warm.