1.In the bowl of a standing mixer, add in order heavy cream, milk, 1 egg, sugar, cake flour, bread flour, yeast, and salt. Using the dough hook attachment, turn on the mixer to “stir.” Let it go for 15 minutes, occasionally stopping the mixer to push the dough together. Cover the bowl with plastic wrap and place in a warm spot for 1 hour. The dough will grow to one and a half its original size.
2.Heat oil in a skillet over medium heat. Add the hot dogs, and allow them to crisp slightly and get some color. Set aside on a plate to cool.
3.Put the dough back under the mixer and with the dough hook and "stir" for 5 minutes to get rid of air bubbles. Dump the dough on a lightly floured surface and cut into 12 equal pieces.
4.Take each piece of dough and roll it into a 10-12 inch rope, keeping the middle thicker than the ends. Roll the rope around a hot dog, tucking in the ends. Place on a parchment-lined baking sheet (keep the buns 2-3 inches apart). Repeat until all the buns are assembled.
5.Cover the baking sheet with plastic wrap and allow the buns to rise in a warm place for another hour. The buns should double in size. After half an hour preheat the oven to 350 degrees F.
6.In a small bowl beat 1 egg with 1 teaspoon water. Brush buns with egg wash, and bake the buns for 7 minutes, then rotate the pan and bake for another 7 minutes.
7.In a bowl add caster sugar and 2 teaspoons hot water, mix until dissolved. Remove from the oven and immediately brush the buns with sugar water. Serve immediately.