Sprinkle yeast and sugar over water in a small bowl and stir to dissolve. Let stand until foamy; about 10 minutes.

With a whisk, combine milk or soymilk, salt, olive oil, and 1 cup flour in a large bowl. Add yeast mixture and beat 2 minutes, or until smooth. Add fresh basil and 4 Tablespoons of cheese. Mix in remaining flour a ¹/₂ cup at a time with a wooden spoon to make a soft dough that just clears the sides of the bowl. Scrape bowl and turn dough out onto a lightly floured surface.

Knead about 5 minutes, adding flour 1 Tablespoon at a time as needed to make a smooth dough. Place in a greased bowl, turn once to grease top, and cover with plastic wrap. Let rise in a warm place until doubled; about 45 minutes to 1 hour.

Punch down dough and turn out onto a lightly floured surface. Divide dough into 12 equal pieces. Form each into a ball, pulling surface tight and pinching seams on bottom. Place in a greased 10-inch pie plate, springform pan, or earthenware baking dish. Cover loosely with plastic wrap and let rise until doubled, abut 20 minutes. Brush with olive oil and sprinkle with the remaining 2 tablespoons of cheese.

Bake in an oven preheated to 400° for 30-40 minutes, or until brown. Slide out of pan to cool on a rack. Pull the rolls apart and serve warm.