Season the pork chops with salt and ½ teaspoon sugar. Heat 1 tablespoon olive oil in a large skillet over medium high. Cook chops, without touching them, until well browned underneath, about 3 minutes. Turn and cook the other side for another minute, slightly browned. Transfer to a plate (they will not be cooked), reduce the heat.
Pour the remaining 1 tablespoon olive oil into the same skillet and add the shallot and garlic. Cook stirring often until softened and just beginning to brown, about 3 minutes.
Add the vinegar and remaining tablespoon of sugar to the skillet. Cook until the vinegar is nearly evaporated, about 3 minutes.
Add rosemary, capers and ½ cup water to the skillet. Season with salt. Reduce the heat as needed to maintain a very low simmer. Add butter and swirl pan continuously until the sauce becomes smooth and emulsified.
Return the pork chops to the skillet arranging darker side up. Simmer gently in sauce until the chops are cooked about 3-5 minutes.
Transfer pork chops to plate and spoon sauce over it.