Combine basil, pine nuts, Asiago, garlic, pepper, and salt in a food processor; pulse until uniformly chopped. Gradually add olive oil until a thick paste forms. Taste and adjust the seasoning.
Melt butter in a medium pan over low heat. Add flour and whisk constantly for 2 minutes. Gradually add milk, stirring constantly. Continue to cook until thickened, about 5 minutes. Stir in Asiago cheese until melted. Stir in pesto sauce until combined.
