Combine ⅓ cup dried tart cherries and ½ cup hot water in a small bowl. Let sit to rehydrate while you cook the pork.

Pat two 1"–1¼"-thick bone-in pork chops (10–12 oz. each) dry with paper towels. Season generously on both sides with kosher salt and freshly ground pepper. Sprinkle 3 Tbsp. all-purpose flour evenly over both sides of each chop; press in gently to coat lightly.

Heat 2 Tbsp. extra-virgin olive oil in a large cast-iron skillet over medium-high. Shake off any excess flour and cook pork chops until deep golden brown underneath, about 5 minutes. Turn and cook until deep golden brown on other side and an instant-read thermometer inserted into the thickest part registers 135°, about 2 minutes, depending on thickness of chops. Transfer to a plate. Let pan cool slightly.

Combine 1 large shallot, thinly sliced, and 2 Tbsp. unsalted butter in same pan and cook over medium heat, stirring often, until shallot is softened, about 2 minutes. Add 2 cups low-sodium chicken broth, 2 Tbsp. Dijon mustard, 1 Tbsp. balsamic vinegar, and reserved cherries with their liquid. Cook, stirring constantly, until slightly reduced, about 5 minutes. Add 2 sprigs rosemary and cook, stirring often, until sauce thickens further and is homogeneous, about 5 minutes. Remove from heat, add remaining 2 Tbsp. unsalted butter, and vigorously stir to combine. Taste sauce and season with salt and pepper.

To serve, nestle pork chops back into sauce in pan and spoon some sauce over.