Heat a large cast-iron pan over medium heat. Add enough olive oil to lightly coat. Add the sausage and cook, breaking it up with the back of a wooden spoon or spatula, until cooked through and no longer pink, 3 to 5 minutes. Remove with a slotted spoon to a plate and set aside.
Add the mushrooms to the pan. Sear until golden brown on all sides, about 3 minutes. Remove from the pan and add to the sausage. Add the onion and cook, stirring and scraping up the brown bits from the bottom of the pan, about 2 minutes. Add about two-thirds of the minced garlic and cook until the onions and garlic are translucent, about 3 more minutes. Add the celery and zucchini and cook, stirring. until just tender, about 3 more minutes. Add the chicken broth and wine. Bring to a boil, then reduce to a simmer and cook until the mixture reduces by half, 5 to 7 minutes. Add the sausage and mushrooms back to the pan and stir in the thyme leaves. Taste and season with salt.
Meanwhile, prepare the chicken: Season the chicken generously with salt. Stuff the center cavity with the lemon half, whole garlic and thyme bundle. Tie the legs together with kitchen twine and tuck the wings under the body to ensure even cooking. Place the chicken on top of the sausage mixture in the pan. Roast until the internal temperature reaches 165 degrees F in the deepest part of the thigh, about 45 minutes. Allow to rest for about 10 minutes.
For the snap peas: Fill a large pot with water and bring to a boil. Add salt until it tastes like sea water. Fill a large bowl with ice and water and add a few hearty pinches of salt. Set aside.
Drop the snap peas into the boiling water and cook until bright green and tender but still slightly crisp, about 1 minute. Immediately transfer the snap peas with a strainer or spider to the ice bath. Strain from the water once cool.
Heat a large skillet over medium heat and coat with olive oil. Add the shallots and cook, stirring, until just tender, about 2 minutes. Add the remaining minced garlic and cook, stirring, until translucent, about 2 minutes more. Add the snap peas. Season with salt and cook, stirring, until the snap peas are hot, about 3 minutes. Add the mint and toss to combine.
Carve the chicken. Transfer the sausage mixture to a serving bowl and serve with the carved chicken on top. Serve the snap peas alongside.