Halve and pit the apricots and set aside.
Pour 2 tablespoons of the olive oil into a medium saute pan over medium-high heat. Saute the onions until lightly browned, about 5 to 7 minutes. Add the apricots to the pan and reduce the heat to medium. Heat the apricots just until their skins begin to soften, for no more than 2 to 3 minutes so the fruit keeps its shape. Add the herbs, 2 tablespoons of the balsamic vinegar and 1/2 teaspoon of salt to the pan and simmer for 3 to 4 minutes. Remove the pan from the heat and allow the mixture to cool.
Meanwhile, prepare 4 (3 to 4-inch) ring molds by lining them with prosciutto slices. Each mold gets 4 slices of prosciutto, which you'll overlap so that the entire mold is covered but open with room for the apricot filling. Once you've draped prosciutto over all 4 molds, spoon in the apricot mixture, filling each mold to the top. Lift the prosciutto slices that are resting open over the mold's edge and drape the slices over the apricot filling, gently patting into place. When you unmold these, the prosciutto should completely encase the apricot filling. Set the filled molds aside while you dress the greens.
In a small bowl, whisk the remaining olive oil, balsamic vinegar, salt, and pepper. Place the greens in a medium bowl, spoon on a little of the vinaigrette - just enough so each leaf is lightly coated - and gently toss. Divide the greens evenly between four individual salad plates. Unmold each prosciutto ring by turning it upside down and over the greens. Drizzle any remaining vinaigrette around the prosciutto ring and top the salad with the shaved sheep's cheese.