Roast the cauliflower. Preheat the oven to 400°F. Toss the cauliflower florets with 2 tbsp olive oil, salt, and pepper. Spread them on a metal sheet pan and roast until deeply golden, about 40 minutes.
Brown the sausage. While the oven preheats, heat the remaining 2 tbsp olive oil in a large pot over medium-high heat. When the oil begins to shimmer, add the andouille and sauté until nicely caramelized. Remove the sausage, set it aside, and turn heat down to medium.
Build the base. Add the diced onion and diced cauliflower core to the same pot. Cook until softened and beginning to brown, about 10-15 min. Stir in the garlic, dried thyme, and smoked paprika and cook until fragrant.
Simmer. Add the chicken stock, Dijon, and fresh thyme sprigs. Bring to a simmer, cover and let sit for about 20 min.
Blend. Remove the thyme sprigs. Using an immersion blender or a stand blender, blend the contents of the pot until smooth. If using a stand blender, return the soup to the pot.
Add the mix-ins. Add the roasted cauliflower, browned sausage, and 2 of the diced apples. Bring to a simmer to heat everything through and slightly soften the apples.
Make the cheese sauce. In a small saucepan, heat the evaporated milk over medium-low. Once it's lightly steaming, add the shredded Gouda a little at a time, stirring gently until completely smooth. Keep the heat low and stir gently, since high heat and vigorous mixing will cause the cheese to clump.
Finish with cheese. Just before serving, reduce the heat to low, just below a simmer. Slowly stir in the cheese sauce until fully combined.
Salt and pepper to taste.
Serve. Top each bowl with fresh thyme and the remaining diced apple.