Place the dried shiitakes into a small bowl and cover with hot water for about 30 minutes to allow them to re-hydrate.
Rinse the chicken to remove any residue and pat dry with paper towels. Rub the mesquite seasoning onto all the chicken parts. Preheat oven to 350 degrees F. In a large saute pan, brown the chicken on all sides and place on a baking sheet. Transfer the chicken to the oven and bake until the juices run clear and the chicken is fork tender, about 35 minutes. Remove from oven and let rest.
While the chicken is in the oven, prepare the tomato demi-glace.
Heat the canola oil in a medium saucepan over medium-high heat and saute the red onion, celery and garlic until they begin to caramelize lightly, being careful not to burn them. Deglaze the pan with the red wine, and when most of the alcohol has evaporated, add the vegetables, stock, tomatoes, parsley leaves, and thyme. Return to a gentle boil and let the liquid reduce by half, then strain into a small bowl.
Strain the mushrooms through paper towels to reserve the soaking liquid. Use 1/2 cup of the mushroom soaking liquid in a small saucepan to make the sauce by whisking in the demi-glace. Heat for a few minutes to let the flavors integrate. Remove from heat and fold in the mushrooms.
Cut the figs in half lengthwise. Arrange the chicken on serving dish, spoon some sauce on top and garnish with figs.