Warm a teaspoon of oil in a large soup pot over medium-high heat, and add the sausage. Cook the sausage until completely browned, breaking it up into crumbles as it cooks. When browned, remove the sausage from the pan with a slotted spoon and set aside.
Drain all but a tablespoon of fat from the pan (add more oil if you don’t have quite a tablespoon). Add the onions and a half teaspoon of salt, and cook until the onions are translucent and beginning to brown, about 5 minutes.
Add the mushrooms and another half teaspoon of salt. Cook until the mushrooms have released all their moisture and are cooked through, another 5 minutes. Add the carrots and cook until they look softened around the edges, another 2-3 minutes. Stir in the garlic and cook until fragrant, 30 seconds.
Pour a cup of chicken stock into the pot and scrape any browned bits from the bottom of the pot. Return the sausage to the pot and add the remaining stock. Bring to a rapid simmer over high heat, then reduce the heat to a medium.
Add the tortellini and cook according to package instructions.
In a medium sized bowl, add the drained beans, ½ cup water, and apple cider vinegar. Puree using an immersion blender (or you can do this in a food processor or countertop blender).
When the tortellini is done, add the greens and stir until wilted. Stir in the pureed beans and heavy cream (or milk). Taste and add salt, pepper, or another splash of cider vinegar as needed to perk up the flavors. Serve.
