You want a broth based broccoli and potato soup?...No milk cheese or cream?
I have never done this because I don't care for potatoes in any soup but potato leek soup...but.
I would cut up 2 small onions...wash and cut into florets and peel and chunk up the stem of 2 stalks of broccoli and 3 or 4 potatoes depending on the size.
I would spread the potatoes on a baking sheet and sprinkle with EVOO and salt, pepper and a bit of garlic powder and roast for 20 to 25 minutes at 375...then I would add the broccoli and turn the potatoes over and roast for another 20 minutes.
Then dump the roasted veggies into your soup pot and add 3 or 4 cups of good chicken broth and simmer for about 15 to 20 minutes.
With just broth and potatoes and broccoli you don't have much to add flavor to the soup...so roasting the veggies would help a lot. Other wise you just have boiled potatoes and broccoli...not very appetising in my opinion.
Linda C

Here are several to choose from.

Here is a link that might be useful: Broccoli Potato Soups