In a large bowl or the bowl of a stand mixer fitted with a dough hook, combine water, sugar, yeast, 1/3 cup olive oil, eggs, salt and flour. Knead in the bowl or with the stand mixer until a soft, smooth ball forms, about 10 minutes by mixer and 15 minutes by hand. Place the ball of dough into a lightly greased bowl, turning to coat then cover with plastic wrap and let rise at warm room temperature until doubled in size, about 45 minutes to 1 hour.

Preheat oven to 325°F. Cut top off of garlic heads. Place in an ovenproof dish and drizzle with 1 tablespoon olive oil and 1 tablespoon water. Cover with foil and bake for 1 hour in preheated oven, uncovering during the last 15 minutes of baking. Set aside.

Line a baking sheet with parchment paper and set aside. Deflate the dough gently and split it into two even pieces. Roll one piece of dough to a rectangle, measuring 22 x 8. Sprinkle dough with half the cheese, basil and tomatoes. Squeeze the roasted garlic from one head and distribute it evenly over the dough.

Starting with one long side, roll into a log lengthwise and pinch edges and seam to seal. Move the log to prepared baking sheet and set seam side down. Using kitchen shears, starting 1/2-inch from one end of the log, snip down the length of the log about 1-inch deep (don't cut all the way through), stopping 1/2-inch from the other end of the dough.Shape the dough into a "S" shape, keeping the cut side up then tuck the ends under the middle of the bread, forming a Figure-8 and pinch the ends together underneath to seal. Repeat entire process with remaining dough. Cover loaves with plastic wrap and let rise at warm room temperature until doubled in size, 45 minutes to 1 hour.

Preheat oven to 350°F. Bake loaves in preheated oven 35-40 minutes, tenting with foil after 20 minutes to prevent over-browning. Remove loaves and transfer to wire racks and cool completely.