Serves 3 to 4
In a Dutch oven, melt butter and oil over medium-high heat. Add green onions, yellow onions, salt, pepper, and thyme and cook, stirring, until vegetables are very soft but not colored, 5 to 6 minutes. Add roasted garlic, stir well to incorporate, and cook until fragrant, 1 to 2 minutes. Add wine, bring to a boil, and cook, stirring, until wine is nearly completely evaporated, about 2 minutes. Add stock and Parmigiano rinds and bring to a boil. Reduce heat and simmer, stirring occasionally, until vegetables are very soft and soup is slightly thickened, 20 to 25 minutes. Add cubes of bread to thicken, as desired, and when bread is soft, puree soup in a blender or using a handheld immersion blender. Add cream, parsley, and lemon juice and simmer for 5 minutes. Adjust seasoning to taste.
To serve, ladle hot soup into bowls and garnish with chopped chives.