Dissolve the salt in 6 cups cold water. Submerse the chicken breasts in this salt-water brine for at least 1 hour in the refrigerator.
Take chicken out of the brine and pat dry with paper towels. Discard brine. Dredge chicken in flour to lightly coat.
Heat a heavy bottomed pan over medium-high heat. Add 2 tablespoons of the olive oil and brown the chicken on both sides, approximately 3 minutes per side. Do not crowd the pan – if the pan is too crowded, the chicken will steam in their juices rather than brown. Once browned, remove chicken from pan and set aside.
Cool down the pan a bit by adding half of the tomatoes and turn down the heat to medium.
Add the remaining 1 tablespoon of olive oil, shallot, garlic, anchovy, and oregano and sauté for about 2 minutes until shallot and garlic are softened.
Add the rest of the tomatoes, wine, and the chicken breasts. Turn heat to high and bring to a boil.
Reduce the heat to simmer, cover and cook over med-low heat for 20 minutes.
Remove the chicken and set aside covered with foil to keep warm.
Increase heat to high, add the tomato paste and the sugar. Reduce the liquid by boiling for approximately 5 minutes until it has reached desired consistency, stirring occasionally.
Remove the sauce from the heat and stir in the olives. Serve immediately; plate one chicken breast per plate, and spoon sauce over it, and garnish with fresh oregano leaves. Great on its own or serve on rice or pasta.