Cut asparagus stalks into 1-inch pieces; reserve tips in a bowl.
Melt 2 tablespoons butter in a Dutch oven over medium heat. Add 1/3 cup chicken broth and leeks; cover and cook until leeks are tender, about 5 minutes. Add asparagus stalks; cover and cook for 3 minutes.
Pour remaining 4 2/3 cups chicken broth into the Dutch oven. Bring to a boil; reduce heat and simmer until asparagus is very tender, 10 to 12 minutes. Add 2 packed cups spinach and sugar; cook until spinach is tender, about 5 minutes.
Puree soup with an immersion blender until very smooth.
Heat remaining 3 tablespoons butter in a small saucepan. Whisk in flour until well blended; cook for 1 minute.
Whisk flour mixture into the pureed soup until smooth and blended. Season soup with salt and white pepper. Bring to a boil; reduce heat and stir in peas. Simmer until peas are barely tender, about 3 minutes. Stir in asparagus tips; simmer until tender, about 5 minutes.
Pour heavy cream into the soup. Cook until heated through but not boiling, 3 to 4 minutes. Stir in remaining spinach; cook until heated through, about 3 minutes.