Preheat the Dutch oven or other large pot over medium-high heat.

Season both sides of the chicken with salt and pepper.

Add two tablespoons of vegan butter and oil to the pot. Brown the chicken in batches, starting with the dark meat. Once both sides are brown, remove from the pot and set aside.

Adjust the heat if needed and add the celery, onions, and carrots. Sauté until the onions are tender, making sure to scrape up all the bits at the bottom off the pot.

Tie the herbs together with string. Add the mushrooms and herb bundle and continue to cook until the mushrooms release their liquid. Add the flour and stir until everything is coated.

Deglaze with the white wine and cook for another minute before adding the chicken stock. Place the chicken back into the pot and bring to a simmer. Cover and cook for about 25 to 30 minutes.

Once the chicken has fully cooked, remove the chicken from the pot and set aside. Whisk together the egg yolks and the oat milk. Slowly add a half cup of the hot pot sauce to the egg mixture while whisking to bring it to temperature. Pour the warm egg mixture into the pot and stir together; this will thicken the sauce and enrich it. Taste the sauce and add salt and pepper to taste.

Place the chicken back into the pot. Add the lemon juice and chopped tarragon, and then serve.