Combine the water, egg, yeast, sugar, flours, salt, and olive oil the the bowl of your stand mixer fitted with the dough hook. Knead until the dough is smooth and elastic. -OR- Mix in a bowl until combined, then finish kneading by hand on a lightly floured surface.
Add the sun-dried tomatoes and knead just until they are evenly distributed in the dough.
Cover the bowl and set aside to rise until the dough has doubled in size, about an hour.
Turn dough out onto a lightly floured work surface. Divide dough in half. Form the first half into a ball, then flatten to form a disk about 10 inches in diameter. You can use a rolling pin, or just stretch and pat and pull it into shape. Try for an even thickness.
Sprinkle the cheese on top of the dough, leaving about an inch around the edge uncovered. Sprinkle the oregano on top of the cheese.
Form the second piece of dough into a disk just like you did with the first piece, and lay that on top of the cheese-covered dough. Pinch all around the edges to seal. Press down on the dough to encourage the cheese to adhere to the dough.
Transfer this to the center of a piece of parchment paper. Use your hands to shape it into an even circle.
Place a small ramekin or small drinking glass in the center of the dough. This will serve as your stopping point when you begin cutting the dough. Make 4 cuts in the dough as though you were cutting an X in the dough if the ramekin wasn't there. Cut all the way through the dough.
Now, make 4 more cuts, centered between the first ones you made. You should now have 8 even strips of dough.
Make 8 more cuts centered between the ones you already made. You will now have 16 even strips of dough.
Take one of the strips and twist it one full turn. Don't pull, just twist. Take the next strip and twist it one full turn in the opposite direction. Continue all around the circle, twisting right and left, alternately, until you have twisted all the dough strips.
Slide the parchment onto a baking sheet and use your hands to even up the circle. Make sure you've got at least an inch of space between the dough and the sides of the baking sheet.
Cover the baking sheet and let the dough rise until doubled, about 30 minutes. If you have a second baking sheet of the same size, you can flip that upside-down and use that as the cover.
While the dough is rising, heat the oven to 350°F.
When the dough has risen, bake bread until it's nicely browned and the exposed cheese is melted.
Remove from the oven and let it cool on a rack for a short while - use caution, the cheese is very hot.
Serve warm or at room temperature with your favorite dipping sauce.