Heat up the milk and water, either in the microwave or in a saucepan, until it is "fingervarm" (finger warm), meaning it should feel warm to the touch but not hot. You don't want to kill the yeast! Add the yeast and stir to dissolve. Let it rest 5-10 minutes until it begins to froth and bubble a little bit. (If your yeast doesn't do anything, it may be expired. Check the expiration date on the package and start over with new yeast!)
Combine the sugar, salt, italian seasoning, tomato paste, and melted butter in a large mixing bowl or the bowl of a stand mixer.
Add in the milk, water, and yeast mixture. Stir to combine.
Add in the flour, stirring with a wooden spoon until the dough starts to come together. Then begin kneading (with the stand mixer, use the dough hook attachment).
Knead for about 10 minutes, adding more flour if the dough is too sticky. You should be left with a soft and smooth dough.
Cover and let your dough rise for about 45 minutes, or until doubled in size.
Once the dough has risen, turn the dough out onto a surface for rolling out.
Roll out the dough into a large rectangle, aiming for about 1/4" thickness.
Sprinkle about ⅔ of the cheese over the dough.
Arrange the pepperoni slices on top of the cheese, creating an even layer.
Sprinkle the remaining ⅓ of the cheese over the top of the pepperoni.
Begin rolling the dough up from the short edge.
Cut the log into 1-1.5 inch (3cm) thick pieces. We divide our dough into 12 pieces.
Arrange the swirls on two baking sheets prepared with parchment paper.
Cover the swirls loosely with cling film or a clean dishtowel and let them rise another 45 minutes. In the meantime, preheat your oven to 350 degrees F (175 degrees C).
Make your egg wash (1 egg whisked together with a splash of water until there are no more gloopy streaks of egg white), then brush over the swirls and bake for about 15 minutes.
Once they are done, let them cool slightly. Enjoy!