In a one-cup measuring cup, combine the yeast, warm water, and 1/2 teaspoon of sugar. Stir and let it sit until it becomes foamy on top, about 5 minutes,.
Meanwhile, in the bowl of a stand mixer add the flour (start with the lower amount), salt, and remaining 2 tablespoons of sugar.
When the yeast is foamy, add it to the dry ingredients in the mixer and turn on low.
With the mixer still running, add the eggs and oil.
Turn up the mixer speed to medium and knead until the dough is smooth and just slightly sticky, about 10 minutes. Only add more flour sparingly by the spoonful if the dough doesn’t come together and release from the bottom of the bowl.
When the dough is smooth, scoop the it up in one hand and oil the bowl with the other. Return the dough to the bowl rolling it around in the oil, cover with plastic wrap, and let it sit at room temperature until doubled in size, approximately 1 1/2 to 2 hours.
When the dough has risen, line a half-sheet baking pan with parchment paper. Roll the dough into a large rectangle (20"x14"). If the filling is wet, drain in a colander. Keep that oil for dipping later!
Evenly sprinkle the cheese over the surface of the dough leaving about one inch clear along one long side of the dough. Dot the tomatoes over the cheese. Tear or slice the peppers and garlic cloves and do the same.
Roll the filled dough from the long side (leaving the plain side for last) into a tight cylinder. When you get to the end, pinch all of the seams to seal along the length and each end.
Using a very sharp knife slice the cylinder from end to end to open up like a book, keeping the cut sides up and touching along their lengths.
While keeping the cut sides up as much as possible, loosely drape the two sides across one another to look like a two-strand braid. Pinch the two ends together at each end to keep them from coming apart during baking.
Slide the baking sheet close to the stuffed dough braid and gently lift it onto the parchment-lined pan. It may get wonky but gently resituate the braid and put any loose filling back on top.
Cover with greased plastic wrap (I use no-stick spray) and let rise for 30 minutes.
Meanwhile, preheat the oven to 350°F with the rack in the center of the oven.
When the dough has risen and the oven has preheated, put the loaf in to bake for 30 minutes or until it’s golden brown and the internal temperature is 190°F.
Remove the loaf from the oven, and cool it on a rack at least one hour before slicing. I leave leftovers on a cutting board covered with a tea towel for a day.