Cook carrots and corn in a pot of about 3 cups boiling water and broth. Leave to boil.
In another soup pot cook onions with butter until transparent, but not browned. Add a dash each of salt, pepper, and dried basil. add flour to pot one tablespoon at a time alternatively with a splash of milk. Make sure to keep stirring so all ingredients mix without clumping. Eventually a thick thick creamy mixture is formed. To this mixture slowly add the carrot and corn soup, about 1/2 cup at a time. Make sure to keep stirring. Once everything in incorporated, let soup boil for a bit more.