Cut off and discard the dark green portions of the leeks and slice off the roots. Quarter the leeks lengthwise and rinse thoroughly under running water; pat dry. Place the leeks, cut side down, in a single layer in the bottom of a square baking dish (see Tip). Sprinkle with the salt and pepper and set aside
In a small bowl, stir together the lemon zest, butter, tarragon, mustard, garlic, and a pinch of salt.
With your fingers, gently loosen the skin from the chicken. Spoon the butter mixture under the skin of each chicken piece and use your fingers to spread the mixture evenly.
Place the chicken pieces, skin sides up, on the leeks in the baking dish. Brush the lemon juice over the chicken and season with salt and pepper.
Bake, uncovered, for 25 minutes.
Remove the chicken from the baking dish.
Add the mushrooms and toss to coat with the juices in the dish.
Place the chicken on top of the vegetables.
Bake for ten minutes more, spooning the juices over the chicken twice, until the internal temperature of the chicken is 180F and the meat is no longer pink.
Check the leeks during the last ten minutes of baking and cover loosely with aluminum foil if they are browning too quickly.
Serve the chicken over the vegetables.
TIP: For easier cleanup, line the baking dish with parchment paper before you add the leeks.