Warm the olive oil in a large pot or Dutch oven over medium heat. Add the sliced smoked beef sausage and sauté for 4 to 5 minutes, stirring occasionally, until nicely browned. Don’t worry if browned bits form on the bottom of the pot. These bits add tons of flavor to the soup. Remove the sausage with a slotted spoon and set aside, leaving the drippings and browned bits in the pot.
Add the diced onion, carrots, and celery to the pot. Sauté for 5 to 6 minutes, stirring occasionally, until the vegetables begin to soften. As they cook, the moisture from the vegetables will help loosen the browned bits from the bottom of the pot.
Stir in the smoked paprika, thyme, and red pepper flakes. Add the minced garlic and cook for about 30 seconds, just until fragrant. Be careful not to burn the garlic.
Stir in the cubed Yukon Gold potatoes and chicken broth. Bring the soup to a boil, then reduce the heat and simmer for 10 to 15 minutes, or until the potatoes are fork tender.
For a completely creamy soup, use an immersion blender to blend the soup directly in the pot until smooth. For a chunkier soup, remove some of the potatoes and vegetables before blending, then blend the remaining soup until creamy and return the reserved vegetables to the pot.
Stir in the heavy cream. Gradually add the shredded cheddar, stirring well after each addition until fully melted and incorporated. Keep the heat low or remove the pot from the heat while adding the cheese to prevent the dairy from scorching or the cheese from clumping.
Return the browned smoked sausage to the pot and add the corn. Simmer over low heat for 5 to 7 minutes, stirring occasionally, until everything is heated through. If the soup is too thick, add a little more chicken broth, milk, or cream to reach your desired consistency.
Taste the soup and season with salt and black pepper as needed. Ladle into bowls and top with shredded cheddar, chopped chives, fresh thyme, and a dollop of Greek yogurt or sour cream. Serve hot with crusty bread, sourdough garlic toast, or your favorite sandwich for dipping.