Put the parsley, capers, the whole garlic clove, the lemon juice, anchovy paste, mustard, 1/2 teaspoon of the salt, and 1/8 teaspoon of the pepper into a food processor or blender. Pulse just to chop, six to eight times. With the machine running, add the 1/2 cup oil in a thin stream to make a slightly coarse puree. Leave this salsa verde in the food processor; if necessary, pulse to re-emulsify just before serving.
In a large frying pan, heat the remaining tablespoon of oil over moderate heat. Season the chicken breasts with the remaining 1/4 teaspoon salt and 1/8 teaspoon pepper and the thyme and put them in the hot pan. Cook the chicken until brown, about 5 minutes. Turn and cook until almost done, about 3 minutes longer. Add the minced garlic and cook for 30 seconds, stirring. Cover the pan, remove from the heat, and let steam 5 minutes. Serve the breasts with their juices and then the salsa verde poured over the top.
Clean, crisp, lightly fruity Italian whites were made for dishes like this. A Soave Classico or Vernaccia di San Gimignano will stand back and let the main course take center stage.