Mix warm water, sugar, and yeast. Let sit 5–10 minutes until foamy.
Combine flour and salt. Add yeast mixture and olive oil. Knead 8–10 minutes until smooth and elastic.
Place in an oiled bowl, cover, and let rise 1–2 hours until doubled.
Mix garlic, rosemary, thyme, mozzarella, and Parmesan for filling.
Punch down dough, roll into 9×14-inch rectangle, spread filling, roll up tightly, slice lengthwise, twist strands, and place in pan.
Cover and let rise 30–45 minutes.
Preheat oven to 375°F (190°C). Bake 25–30 minutes until golden and hollow-sounding.
Cool 10 minutes in pan, then transfer to rack. Rest 15 minutes before slicing.
