If you have time, leave the chicken uncovered in the refrigerator for a couple of hours, even overnight or all day. If not, proceed. :)
Preheat oven to 400°F.
Heat a large skillet over medium-high heat. Season the chicken with salt and freshly ground pepper, then place them skin side down in the skillet. Cook until they are well-browned, then turn them to the other side and do the same. They won't be done through. Remove to a platter and set aside.
Cook the Andouille slices the same way and put them on the platter with the chicken.
If there's too much fat in the skillet, you can remove some, then sauté the sliced mushrooms till they have released most of their moisture and are browning. Then add the garlic and white wine. Bring to a boil, scraping up any browned bits. Put the mushrooms on the platter with the chicken and Andouille.
Wipe the pan to remove any liquid, then add a little oil and the tomatoes. Stir occasionally until the tomatoes start to blister and brown.
Return the chicken, Andouille, and mushrooms to the skillet. Put it in a 400°F oven for 30 minutes or until cooked to an internal temperature of about 170°F when tested in the thickest part of the thigh.
To serve: Since you've got the meat and veggies all in one skillet, you probably only need a starch like rice, mashed or roasted potatoes, or my favorite ... a creamy polenta. Or even just crunchy bread! If you're not into starches, you could serve some sautéed greens or squash and that would be delicious!