In a large bowl or stand mixer bowl, combine yeast and warm water. Let sit 5–10 minutes until dissolved and lightly foamy.
Whisk milk, pesto, sugar, 1 tablespoon olive oil, salt, and egg. Add to yeast mixture and stir to combine.
Add 3 cups flour and Parmesan; mix until a shaggy dough forms.
Knead until smooth and soft, 8–10 minutes with a mixer or 10–12 minutes by hand, adding remaining flour 1 tablespoon at a time only if needed.
Place dough in a lightly oiled bowl, cover, and let rise 1–1½ hours, until doubled.
Brush a 12-inch cast-iron or oven-safe skillet with 2 tablespoons olive oil. Preheat oven to 350°F.
Punch down dough and divide into 12 pieces. Roll into balls and arrange in skillet. Sprinkle Asiago between rolls, spoon remaining pesto into gaps, and brush tops with remaining 1 tablespoon olive oil.
Cover loosely and let rise 30 minutes, until puffy.
Bake 30 minutes, until golden and cheese is bubbling. Finish with Parmesan and basil. Serve warm.