Pat the pork chops dry. Trim any excess fat - you don't need or want to trim all the fat, just any large chunks or loose bits.
Season both sides with salt (roughly1/4 teaspoon total) and freshly ground black pepper.
Heat the oil in a large skillet over medium-high heat.
Add the chops so they're not touching each other. Try to have the fat side toward the middle of the pan where it's hotter. Let them cook for about 4-6 minutes per side, until they reach 145°, then remove the chops to a plate.
Add the shallot, mushrooms and 1/2 teaspoon salt to the pan and cook, stirring occasionally, for 8-10 minutes. Try to get the mushrooms in a single layer - the pan might be too crowded at first but they'll shrink as they cook. Then stir them only occasionally. You really want them to brown so you don't want them to stir too often.
Meanwhile, whisk together the chicken broth and arrowroot.
Once the mushrooms are browned and the shallot is soft, stir in the garlic, anchovy paste or tomato paste, thyme and rosemary.
Then stir in the broth/arrowroot mixture and use a wooden spoon or spatula to scrape up the brown bits on the bottom of the pan. (This will add flavor and also make the pan easier to clean.) Let the sauce simmer for 2-3minutes, until somewhat thickened.
Stir in the coconut milk and vinegar.
Add the chops back in the sauce, along with any juices accumulated on the plate. Turn the chops in the sauce to coat them evenly, taste the sauce to adjust any seasonings, then serve!