Combine the cherries and red wine in a small bowl. Set aside to macerate until the cherries have plumped, about 2 to 3 hours, depending on how dry the cherries are. Drain the cherries and put them in a small bowl, reserving the wine as well.
For the herbed compound butter, combine the butter, herbs, and salt in a small bowl and stir until thoroughly blended. For the hotel butter, combine the butter, juices, garlic, salt, and pepper in a small bowl and stir until thoroughly blended (if you have a hard time incorporating all the juice into the butter, don’t worry; simply pour extra juice into the skillet when adding the butter). Refrigerate until needed.
Heat a large, heavy skillet over medium-high heat. Season the pork with salt and pepper. When the skillet is hot, add the oil, then add the pork and brown well, about 2 minutes per side. Reduce the heat to medium and add the herbed compound butter and the rosemary sprig. Use a spoon to baste the meat a few times as it cooks for 2 minutes longer, turning the pork once.
Add 1/2 cup of the reserved maceration wine to the skillet and baste the meat with it, cooking until the wine is nearly reduced to a syrupy consistency, 2 to 3 minutes. Turn the pork and add the shallot and garlic, stirring for about 1 minute. Add the demiglace and reduce for about 2 minutes, turning the pork halfway. Add the hotel butter, stirring until it is fully melted. Check the pork for doneness, minimum 145°F in the center or just light pink when cut into, cooking a bit longer if needed (add a bit more demiglace or stock if not much liquid left). Transfer the pork to a plate, cover, and set aside for a few minutes.
Add the drained cherries to the skillet with any accumulated soaking wine in the bowl and simmer gently to heat through for 1 to 2 minutes. The sauce should have a nice coating consistency, simmer a bit longer if needed. Remove the rosemary and taste the sauce for seasoning, adding more salt or pepper to taste.
To serve, cut each of the pork chops into slices and arrange them on warmed plates, or serve the pork chops whole. Spoon the cherries and sauce over, and serve right away.