1. Cut the goat cheese into 6 equal rounds and gently shape each piece into a 2 1/2-inch medallion.
In a small bowl, combine 1/2 teaspoon of the garlic with 1/2 teaspoon of the rosemary and season with salt and pepper.
Sprinkle both sides of the goat cheese with the garlic mixture.
2. Put the flour, egg and bread crumbs into 3 separate shallow bowls. Dredge the goat-cheese medallions in the flour, then dip in the egg and coat with the bread crumbs.
Transfer the breaded medallions onto a platter lined with wax paper and refrigerate until chilled.
3. In a small bowl, whisk 2 teaspoons of the vinegar with the walnut oil and season with salt and pepper.
4. In a small saucepan, combine the remaining 1 tablespoon plus 1 teaspoon vinegar with the remaining 1/2 teaspoon garlic and 1/4 teaspoon rosemary.
Add the sugar and stir over moderately high heat until it dissolves. Add the plums and cook, stirring occasionally, until softened, 5 to 8 minutes.
Transfer the plum mixture to a blender along with 1 tablespoon of the olive oil and the water and blend until smooth. Return the plum sauce to the saucepan and keep warm.
5. In a medium nonstick skillet, heat the remaining 1/4 cup of olive oil until shimmering. Add the goat-cheese medallions and cook over moderately high heat until golden and crisp, about 2 minutes per side.
6. In a large bowl, toss the field greens with the walnut vinaigrette and olives and season with salt and pepper.
Spoon the plum sauce onto 6 plates. Set 1 goat-cheese medallion in the center of each plate and arrange a slice of the prosciutto alongside with a little of the field greens.
Sprinkle with the chives and serve immediately.