Heat a large soup pot over medium-high heat. Crumble sausage into pot; cook and stir until browned, about 10 minutes. Drain and discard grease.
Stir half-and-half, potatoes, chicken broth, milk, onion, oregano, and red pepper flakes into sausage, bring to a boil, and reduce heat to low. Simmer until potatoes are tender, about 30 minutes. Season with black pepper; stir kale into soup. Simmer until kale is tender, 10 to 15 more minutes.