Melt butter in a saucepan over medium heat; stir flour, basil, pepper, and salt into butter and pour in chicken broth. Cook and stir broth mixture until thickened and boiling, 5 to 7 minutes. Stir potato, carrot, and green onions into broth mixture; bring to a boil. Reduce heat and simmer until potatoes are mostly tender, 10 to 15 minutes.
Stir half-and-half and corn into broth mixture; cook and stir until chowder is heated through, about 5 more minutes.