Asparagus-Pesto Lasagna

1/3 cup all-purpose flour

3 

1/2 cups low-fat milk , divided

6 Tbs. pesto *, or more to taste

2 Tbs. grated Parmesan cheese , plus additional for garnish, optional

1 tsp. salt

1/4 tsp. ground black pepper

2 tsp. olive oil

1 

1/4 lb. asparagus spears, tips cut off and reserved, spears trimmed and chopped into 1/4-inch pieces

1 clove garlic, minced (about 1 tsp.)

16 no-cook lasagna noodles (9 oz.)

2 cups shredded Fontina or part-skim mozzarella cheese (8 oz.), divided