Preheat the oven to 425°.
Heat a cast-iron skillet over medium-high. Season the chops with salt and pepper. Add a fat drizzle of oil to the pan; swirl to coat the bottom of the skillet. Cook the chops until browned on both sides, then hold with tongs to brown the edges, 6 to 8 minutes total; transfer to a plate.
Reduce the heat a bit, add the butter to the skillet and let it melt. Add the sage; cook until crispy and the butter browns, about 2 minutes; transfer to paper towels. Add the shallots and garlic to the butter and cook, stirring often, until aromatic, 1 to 2 minutes. Stir in the vinegar, then the stock. Add the beans; season.
Return the chops to the skillet. Roast in the oven until an instant-read thermometer inserted horizontally into the chops registers 140°, 10 to 12 minutes.
In a large bowl, toss the arugula, fennel and Parm; dress with the lemon juice and olive oil. Season with salt.
Arrange the chops on plates. Top with the bean mixture and garnish with the fried sage leaves. Serve with the salad.