4 cups reduced-sodium chicken broth, “no-chicken” broth (see Note) or vegetable broth
½ cup half-and-half (optional)
½ teaspoon salt
Freshly ground pepper to taste
Heat butter and oil in a Dutch oven over medium heat until the butter melts. Add onion and celery; cook, stirring occasionally, until softened, 4 to 6 minutes. Add garlic and thyme (or parsley); cook, stirring, until fragrant, about 10 seconds.
Stir in broccoli. Add water and broth; bring to a lively simmer over high heat. Reduce heat to maintain a lively simmer and cook until very tender, about 8 minutes.
Puree the soup in batches in a blender until smooth. (Use caution when pureeing hot liquids.) Stir in half-and-half (if using), salt and pepper.
To make ahead: Cover and refrigerate for up to 4 days or freeze for up to 3 months.
Note: Chicken-flavored broth, a vegetarian broth despite its name, is preferable to vegetable broth in some recipes for its hearty, rich flavor. Sometimes called “No-Chicken Broth,” it can be found with the soups in the natural-foods section of most supermarkets.