Place a large pot of water on to boil for gnocchi.
Heat a pot with 1 tablespoon extra-virgin olive oil, a turn of the pan, over medium-high heat. Add chicken, season with salt and pepper, and lightly brown. Remove to a plate and add another tablespoon of olive oil and 2 tablespoons butter and heat. Add mushrooms, carrots, celery, and bay leaf, season with salt and pepper. Soften 5 minutes, sprinkle in flour and cook 2 minutes. Whisk in 2 1/2 cups stock, slide the chicken back into the pot and simmer to allow sauce to thicken.
Place the remaining 1/2 cup stock in food processor with basil, parsley, garlic, pine nuts and cheese, salt and pepper. Turn processor on and pour in the remaining few tablespoons oil.
Melt 2 tablespoons butter in a skillet over medium heat with the zest of 1 lemon.
Salt cooking water and drop gnocchi in the water, cook until the gnocchi floats, drain and add to the lemon butter and season with salt and pepper, to taste. Toss the gnocchi 2 to 3 minutes in lemon butter until light golden at edges.
Turn heat off chicken and stir in pesto sauce. Serve the chicken in shallow bowls and top with lemon butter dumplings.