In a large bowl, whisk together 3 ½ cups flour, yeast, sugar, and salt.
Then, add in the water.
Using a spatula, fold until mostly combined. Sprinkle a little more flour if needed to not be sticky and form a ball.
Once ball is formed, grease the same bowl mixed in and place the dough ball back into the bowl. *If you are oil free, then just make sure sides of bowl are floured, or use a silicone bowl to easily get it out Cover with a towel or plastic wrap and let sit for 60-90 minutes in a warm, draft free place until doubled.
While the dough is rising, make the pesto by adding everything from the filling into a food processor except the mozzarella cheese. Blend, scraping down as needed until well combined.
Once doubled in size, pull out the dough and lay it out on a lightly floured surface.
Using a rolling pin, roll dough into an approximately 18×12″ rectangle.
Spread the pesto out evenly across the dough. Sprinkle the mozzarella evenly across the dough.
Carefully roll into log from the longest side up. One completely rolled, place seam side down. Cut into 12 even sized pieces. *It's okay if some of the sauce comes out, but try to tuck and roll as you go along. Place into a greased jumbo muffin tins. Let rise for 20 minutes.
Preheat oven to 375 degrees F.
Once heated, place dish inside and bake in oven for 30-35 minutes or until dough is cooked inside.
Remove rolls from oven when done and brush with vegan butter. Add any additional toppings.
Store any leftovers in an air tight container.