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Whisk together shallot, vinegar, whole grain mustard, granulated sugar, and Dijon mustard in a medium bowl. Gradually stream in ½ cup oil, whisking vigorously until emulsified. Whisk in oregano and season with salt and pepper. Set vinaigrette aside.
Prepare a grill for medium-high heat; oil grate. Grill plums, cut side down, until charred and fruit releases easily from grill, about 2 minutes. Transfer to a cutting board and let cool.
Season pork chops all over with salt and pepper and sprinkle with brown sugar and paprika. Grill, turning occasionally, until deeply browned and an instant-read thermometer inserted near the bone registers 140° (internal temperature should climb to 145° as chops rest), 10–12 minutes. Transfer to a platter and let rest 10 minutes.
Meanwhile, cut plum halves into 2 or 3 wedges each and add to reserved vinaigrette; toss gently to coat. Season plum mostarda with salt and pepper.
To serve, toss arugula with some plum mostarda in a large bowl to coat; transfer to plates. Spoon more plum mostarda over pork chops; serve chops with arugula salad alongside.