In a large Dutch oven, heat the butter and olive oil over medium heat. Add the onion and cook for 5 to 6 minutes, or until softened.
Stir in the garlic and cook for 30 seconds.
Add the zucchini and broccoli and cook for 5 to 6 minutes, stirring occasionally, until the vegetables begin to soften and the zucchini develops a little color.
Stir in the potatoes, Italian seasoning, smoked paprika, salt, and pepper. Pour in the chicken broth and bring the soup to a boil. Reduce the heat to a simmer and cook for 20 to 25 minutes, or until the potatoes are very tender.
Using an immersion blender, carefully blend the soup until smooth.
Reduce the heat to low and stir in the whole milk or half-and-half, heavy cream, Dijon mustard, and cheddar cheese. Heat gently, stirring until the cheese is melted, but do not allow the soup to boil.
Stir in the lemon juice and fresh parsley before serving.