Heat 1 tablespoon olive oil in a large pot or Dutch oven over medium-high heat. Add the hot Italian sausage, breaking it up with a wooden spoon, and cook until browned and crispy, about 6-8 minutes. Use a slotted spoon to transfer the sausage to a plate, leaving the rendered fat in the pot.
Add the chopped onion, sliced carrots, and minced garlic to the pot. Sauté for 3-4 minutes, until the onion is translucent and the carrots begin to soften. Add the 1/2 teaspoon crushed red pepper and stir for 30 seconds, just until fragrant.
Pour in the 1/2 cup white wine, scraping up any browned bits from the bottom of the pot. Let the wine simmer for 1-2 minutes, reducing slightly.
Add the browned sausage back to the pot, along with the Parmigiano Reggiano rind, 4 cups chicken broth, drained beans, 1 teaspoon kosher salt, 1/2 teaspoon black pepper, thyme, and bay leaves. Bring the soup to a simmer, then reduce the heat to medium-low and cook for 20-25 minutes, allowing the flavors to meld.
Stir in the baby spinach or kale, letting it wilt into the soup for 2-3 minutes.
Stir in the juice of 1/2 lemon, then taste and adjust the seasoning as needed.
Ladle the soup into bowls, garnish with freshly chopped parsley, and top with a generous sprinkle of grated Parmigiano Reggiano. Serve with crusty bread for dipping.