Drizzle some olive oil into a heavy-bottomed pot set over medium heat. Add the onion, garlic and some salt and pepper and cook until tender, 3 to 4 minutes. Pour in the chicken stock and half-and-half and bring to a simmer. Add the broccoli and cook until tender. Using an immersion blender, puree the soup until smooth. Stir in the herbed cheese and the Monterey Jack and let melt. Serve warm.