Slice andouille sausage into small discs and sauté in a large dutch oven or stock pot on medium heat.
Cook for 4-5 minutes until golden and caramelized on both sides. Transfer to a plate to rest.
Add olive oil to the sausage fat. Add chopped onions and diced garlic and stir to combine.
Cook until translucent and soft, about 3-4 minutes. Add paprika and stir to combine.
Peel and chop potatoes into small chunks and add to the pot. Stir to combine and cook for 1-2 minutes.
Lower heat to low-medium and add chicken stock. Cook until soup is at a low boil.
Continue to boil for 20-30 minutes, or until the potatoes are fork-tender and easy to blend.
Blend the soup until creamy and thickened on high speed with an immersion blender. Add chicken stock to thin to your desired consistency if needed.
Add 3/4 the cooked sausage back to the pot and stir. Continue to cook for 5-10 minutes to infuse with the sausage flavor.
Chop kale into thin ribbons and add to the soup just before serving. Stir to combine and season to taste with salt and pepper.