Fry onions and then cover with chicken broth, simmer for 15-20 minutes in a covered small saucepan, until onions are soft.
Heat milk in microwave for 1-2 minutes. In a separate medium-sized stockpot, melt butter then add flour, salt and pepper. Cook together, stirring constantly, for 2 minutes, then whisk in warmed milk. Stir and heat together on medium heat until soup starts to thicken, about 5-8 minutes.
Add warmed chicken broth and continue to stir until well blended. Add cheeses and stir until completely melted. Stir in steamed broccoli.