In a 4 to 5 quart Dutch oven or stock pot, heat the butter and olive oil until melted and shimmering.
Add the chopped vegetables and saute until soft (5-8 minutes). Add the minced garlic and continue sauteing until fragrant.
Scoot the vegetables to one side of the pan, and add the sausage to the cleared area. Stir the sausage pieces in place until the edges are golden (3 to 4 minutes). Mix into the vegetables.
Scoot the sausage/vegetables to the side once again, and add a glug of oil to the cleared side. Add the tomato paste, seasonings, and a big pinch of salt to the oil, and stir to create a fragrant paste. Stir into the vegetables.
Sprinkle the flour over the vegetables and mix well. The vegetables will be thick and pasty. Add a splash of broth to loosen things up. Stir for several minutes, to cook the "raw" out of the flour.
If there is flour paste stuck to the bottom of the pot, use broth or a splash of white wine to deglaze the pan (white wine is very effective, but optional).
Add the remaining broth, tomato sauce, tomatoes, and lentils, and bring the soup to a light boil. Reduce heat to maintain a gentle simmer, cover partially with a lid, and let the soup cook for 40 minutes. Check occasionally, stirring well through. If too thick, add a small amount of water, and continue cooking.
When the lentils are cooked through (they should be plump and al dente but tender), turn off the heat and stir in the spinach. Taste, and add more kosher salt if needed.