In a frying pan, melt the butter over medium-low heat. Add the shallot and the rosemary sprig and cook for about five minutes, until the shallot has softened. (It should smell heavenly.)
Meanwhile, dry the livers and season with salt and pepper. After the shallot has softened, add the livers to the pan and cook for about three minutes per side, or until they're just pink in the middle. Take the pan off the heat and transfer the livers and shallots to a food processor. (Discard the rosemary.) Add the capers to the liver and shallot mixture.
Return the pan to the heat and add the vermouth. Bring the vermouth to a boil and deglaze the pan, scraping the brown bits off the bottom. When the vermouth has reduced by half, add 1/2 teaspoon coarse salt and the vinegar. Stir and then add to the food processor along with the livers.
Pulse the food processor until you have a coarse paste. Taste for seasoning and serve.