Select the sauté function on your Instant Pot, Ninja Foodi, or slow cooker.
Add the butter and olive oil to the inner liner and let them melt. Once hot, add the onion, carrot, celery, and mushrooms. Sauté over medium heat, stirring occasionally, until the vegetables are softened.
Sprinkle the flour evenly over the vegetables and stir well. Let it cook briefly, stirring constantly, just until the flour is incorporated. Do not let it brown.
Slowly pour in the chicken broth, stirring constantly to avoid lumps. Add the tarragon, thyme, white pepper, black pepper, hot pepper sauce, and parsley. Stir to combine.
Cover and switch to the slow cooker function. Cook on low for 6 hours.
When ready to finish the soup, stir in the half-and-half, lemon juice, and cooked chicken. Switch back to the sauté function and bring the soup to a gentle simmer. Cook for about 10 minutes, stirring occasionally.
Taste and adjust seasoning as needed, then serve warm.