Optional: Condimentssliced tomato, lettuce, and onions
In a large bowl (for best results use a stand mixer and dough hook) combine 3 cups of flour, salt, sugar, and yeast.
Heat the water and milk until very warm – 120 F to 130 F – in the microwave or in a small pot on medium heat.
Add egg and warm milk mixture to dry ingredients in mixing bowl. Beat 2 minutes on low.
With mixer running, slowly add oil. Beat until combined.
Gradually add remaining flour until dough comes cleanly away from the sides of the bowl. Knead by machine for 5 minutes, or by hand on lightly floured board for 6-7 minutes. Dough should be soft and elastic. If it is sticky, add a little more flour. Cover and allow dough to rest for 20 minutes.
Working with half of the dough at a time on a very lightly floured surface, roll dough out 1/3 inch to 1/2 inch thick. Cut shapes with 3-inch star cookie cutter and place on parchment-covered baking sheets, at least 1 inch apart. Set scraps aside as you work and roll them all at once. If the dough gets too tough to roll, form a ball, cover, and let rest for 20 minutes, then try again.
Cover and allow stars to rise until almost doubled – about an hour.
Heat oven to 400 F.
Add 1 teaspoon water to reserved egg white and whisk until foamy. Brush onto tops of stars and sprinkle with sesame seeds if desired. Bake for approximately 8 minutes, or until light golden brown.
Move to cooling racks. Slice horizontally once cool.
Form into round patties about 3-1/2 inches across (to allow for shrinkage). If you want to make stars, roll meat out between sheets of parchment, about 1/3-inch thick, and cut with the same cookie cutter used for the buns. If your meat has a higher percentage of fat, use a sharp knife to cut out the stars using the cookie cutter as a guide and cutting 1/4-inch outside the cookie cutter, making a larger star to allow for shrinkage.
Place meat stars on a hot griddle. When one side is browned, flip it over, add a slice of cheese, and top with the top half of a star bun. This will warm and soften the bun. When the bottom is done, slide it onto the bottom half of a bun. Stick a toothpick through the center to hold the slider together and serve.