Heat a large, deep skillet or a medium soup pot over medium-high heat. Add the EVOO and butter. Melt the butter into the EVOO, then add the potatoes, onion, celery and carrots, adding them to the pot as you chop them. Add the bay leaf and season the veggies with salt, pepper and poultry seasoning, then cook for 5-6 minutes to soften them a bit. Add the flour and cook another minute, then whisk in the wine and cook off a minute more. Add the stock and put a lid on the pan or pot and raise the heat to bring to a quick boil.
Slide in the chicken and cook for five minutes. Stir in the asparagus and cook for three minutes more. Turn off the heat and add the peas and tarragon. Stir to combine and adjust the salt and pepper, to your taste.
Ladle into bowls and serve with crusty bread for mopping.