In a 5- to 6-quart pan over medium-high heat, frequently stir onions, garlic, and butter until onions are limp, 10 to 12 minutes.
Add broth and potatoes. Cover, bring to a boil over high heat, then reduce heat and simmer until potatoes mash easily when pressed, about 20 minutes. Stir in peas and tarragon; add cayenne to taste.
In a blender, whirl soup, a portion at a time, until very smooth. Return to pan and stir over high heat until hot, 2 to 3 minutes.
Ladle into bowls and season to taste with salt and pepper.