Preheat oven 200 degrees F.
Place flour in a shallow bowl; season with salt and pepper. Dredge each chicken paillard in flour, shaking off excess. Strain reconstituted porcinis, saving liquid. Chop porcinis and set aside.
In a large skillet, heat 2 tablespoons oil over medium-high heat. Add chicken and cook until golden brown, about 4 to 5 minutes per side. Transfer to an ovenproof plate and keep warm in the oven.
Add remaining teaspoon of oil to the pan over medium heat, add shallots and cook 1 to 2 minutes. Turn heat to high and add the balsamic vinegar, porcini liquid and apple cider. Stir to incorporate any browned bits in the pan and bring to a boil.
Once boiling, reduce to a simmer over medium-low heat and add mushrooms and thyme leaves to sauce. Allow to simmer until sauce has reduced and thickened slightly, about 5 minutes.
To serve, place one chicken paillard on each plate and top with some of the porcini-cider sauce. Garnish with additional thyme sprigs before serving.