Melt 4 tablespoons of the butter over medium heat in a Dutch oven or other large, heavy pot. Just as the foam subsides, add the flour, stirring constantly with a wooden spoon or whisk to prevent lumps. Pour in the chicken stock and continue to stir well. Bring the mixture to a simmer, and then add the barley and the salt. Reduce the heat to medium-low flame, cover, and cook for about 25 to 30 minutes, or until the barley is tender.
In a large mixing bowl, blend together the yogurt and egg yolks until smooth.
After the barley has softened, lower the heat and gradually whisk the yogurt mixture into the barley and cook just until it is ready to boil.
For the finishing touch, melt the remaining 2 tablespoons of butter in a saute pan. When the butter is foamy, mix in the mint and paprika. Let this turn golden and stir it into the soup. Serve the soup piping hot.