Special equipment: an instant-read thermometer, 1 number 3 pastry tip and a small piping bag
Preheat the oven to 400 degrees F.
In a small bowl, whisk together 2 of the eggs and 1 tablespoon water until the whites break down and the mixture is thin and runny. Set the egg wash aside for later use.
In a medium bowl, whisk together the 3/4 cup warm water, sugar and yeast until dissolved and bubbly. Using a wooden spoon, add the butter, 2 cups of the flour and the remaining 2 eggs and mix until a dough starts to form.
Place the dough in the bowl of a stand mixer fitted with the dough hook and mix on low speed. Mix the salt into the remaining 2 cups flour and gradually add the flour to the dough with the mixer still on low speed. Continue to mix until the dough is smooth, 3 to 5 minutes. Place the dough in a lightly greased bowl to rest at room temperature until it doubles in size, 1 to 2 hours.
Punch down the dough and divide it in half. On a lightly floured work surface, roll out 1 portion of the dough to about 1/4 inch thick with a rolling pin or dowel. Cut the dough into 1/4- to 1/2-inch-wide strips, using a knife or pizza wheel. Roll out the remaining dough and repeat.
Wrap the hot dogs with the strips of dough starting at one end and wrapping to the other end, leaving a small space for the eyes, overlapping the strips if desired.
Place the wrapped hot dogs on a parchment-lined baking sheet about 2 inches apart and brush with the reserved egg wash. Bake until golden brown, 12 to 17 minutes.
Cut the olives into quarters from end to end using a small paring knife. Remove some of the interior meat from the olives using the same knife. Using the small end of a small piping tip, punch out the eyes for the hot dogs.
Place the mayonnaise in a small piping bag. When the dogs cool slightly, pipe the mayonnaise into 2 small circles connected in the center. Place the olive pieces in the mayonnaise to create the finished eyes. Repeat for the remaining hot dogs. Serve the mummy hot dogs warm with ketchup and mustard.