Heat a large stock pot over medium-low heat. Add olive oil, onions, sausage and potatoes. Saute until onion is translucent and soft, about 5 minutes. Season with 1/2 teaspoon kosher salt.
Add stock to the mixture and bring to a simmer. Cover and cook for 10 to 15 minutes until potatoes are soft.
Meanwhile, heat a medium saucepan over medium heat. Add butter and melt. Whisk in flour and cook for 2 minutes. Whisk in half the milk and stir until slightly thickened. Add additional milk and cook until slightly thickened. Remove from heat and stir in cheddar until melted.
After potatoes have softened, remove the pot from heat. Slowly whisk in the cheese sauce. Divide into bowls and top with chopped green onion.