Preheat the oven to 375 degrees F. Pat the chicken pieces dry with a paper towel and sprinkle all over with salt and pepper. Set a turkey roaster pan over 2 burners on medium heat and add the canola oil. Brown the chicken, skin-side down first, 3 to 4 minutes, then turn and brown on the other side, 2 to 3 minutes longer. Remove from the pan and set aside.
Add a little more of the oil to the pan on high heat and add the mushrooms, garlic and onions. Season with salt and cook just long enough to get some color on the mushrooms, 2 to 3 minutes. Stir in the tomato paste and thyme and cook about 1 minute longer. Place the chicken pieces back into the pan skin-side up and spread them out evenly, pushing them down into the mushrooms and onions. Add the marsala, scraping up the brown bits from the bottom of the pan, and add the chicken broth. Place the whole pan in the oven and cook until the tops of the chicken are golden brown, 30 to 35 minutes.
Use tongs to remove the chicken from the pan and transfer onto a serving platter and keep warm. Set the roasting pan back over the burner on high heat. Bring the contents to a boil and cook the liquid until it is reduced and the flavors become concentrated, 3 to 4 minutes. Shut off the heat, add the butter cubes and stir gently to melt the butter evenly. This will thicken the marsala sauce and give it a rich flavor. Taste and season with salt and pepper. Add a squeeze of the lemon juice and fold in the parsley. Pour over the chicken and serve with Chardonnay.