Apple butter and figs add sweetness to this filling dinner dish.
In a small skillet over medium heat or on a tray in a toaster oven set at 350°, toast the nuts until golden, stirring often, 6 to 8 minutes in the skillet or 7 to 8 minutes in the oven. Let cool.
Cut each chicken breast horizontally, almost in half. Open like a book and place between 2 sheets of plastic wrap or parchment paper. Using the flat side of a mallet, pound the chicken to 1/8 to 1/4 inch thick; season with salt and pepper.
In a large cast-iron or nonstick skillet, heat the olive oil, two turns of the pan, over medium-high. Working in batches, cook the cutlets until golden brown and cooked through, about 3 minutes per side. Transfer to a plate and cover with foil.
Add the butter to the skillet. When it foams, add the shallot and garlic; season with salt and pepper. Cook, stirring often, until the shallot softens, a minute or two. Sprinkle in the flour and whisk until blended. Whisk in the calvados, then the stock. Cook, stirring often, until the sauce thickens a bit, about 4 minutes. Stir in the apple butter, then the Worcestershire. Remove the sauce from heat.
In a large bowl, add the arugula and lemon juice. Drizzle with EVOO. Season the arugula with salt; toss until coated.
Add the cutlets to the sauce and turn until coated. Arrange the cutlets on plates. Spoon any remaining sauce on top. Arrange the arugula and figs on top of the chicken with the cutlets peeking out. Scatter the nuts and cheese on top.