Heat 1/4 cup butter and oil in a large pot over medium heat until butter melts, 1 to 2 minutes. Add onion; cook and stir until soft, about 5 minutes. Add remaining 1/4 cup butter, portobello mushrooms, baby bella mushrooms, and oyster mushrooms, and tarragon; cook and stir until tender, 5 to 7 minutes.
Stir chicken broth, potato, thyme, and oregano into the pot. Bring to a boil; let soup simmer until potato is tender, 15 to 20 minutes.
Remove soup from heat and puree with an immersion blender until smooth.
Return soup to heat and simmer until heated through, about 5 minutes. Stir in heavy cream. Season with salt and pepper.