In a large pot, melt the butter over medium-heat. Add the onion and garlic and season with salt and pepper. Cook for 5 minutes, or just until the vegetables begin to soften. Add the flour and cook for 1 minute. Stir in the broth and rice. Bring it to a boil, reduce the heat and simmer for 14 minutes, or until the rice is almost tender. Add the broccoli and the chicken and cook for 6 minutes, or until the broccoli is tender but still beautifully bright green. Reduce the heat to low. Add the half and half and cream cheese chunks. Stir until the cream cheese has melted into the soup. Stir in the lemon juice and season the soup to taste with salt and pepper, adding more broth if the soup seems too thick.
To serve, ladle the soup into six bowls. Garnish each bowl with ¼ cup shredded cheddar and 2 or 3 coarsely crumbled crackers.
Notes
If you'd like to make this soup vegetarian, leave out the chicken. Add 4 peeled and chopped carrots with the onion. Substitute vegetable broth for the chicken broth.