Scrub the potatoes and cut in about 1" chunks.
In a stock pot cover the potatoes with water.
Bring to a boil and simmer over medium heat for 12 minutes. Drain.
Steam the broccoli for 10 minutes. Set aside.
In the same stock pot, melt the dairy-free butter over medium heat.
Stir in flour until smooth. Watch closely.
Gradually stir in non-dairy milk and vegetable broth over high heat, and bring to a boil, stirring constantly. Turn down the heat to medium. Cook for about 10 minutes.
If you would like the soup a little thinner then you can add more vegetable broth, ¼ cup at a time.
Gently stir in potatoes, broccoli, and onion. Add the dried tarragon.
Heat through over medium so that the vegetables get hot and completely cooked, about another 20 minutes. The potatoes and broccoli should be able to be pierced easily with a fork.
Add salt and pepper. Serve hot.
Sprinkle a little fresh tarragon leaves and pepper on top, if you like.