Heat the oven to 400°F. Put a large pot of salted water on to boil.
Put the tomato wedges in a colander set over a bowl. Sprinkle with with 1 Tbs. salt, toss, and let stand for 30 min.
Slice the prosciutto crosswise into 1/2-inch strips. Arrange on a baking sheet in a single layer and bake until crisp and light golden, about 10 min. Set aside.
Meanwhile, when the water comes to a boil, add the beans and cook until tender, 4 to 6 minutes. Drain and let cool.
In a small bowl, whisk together the chopped savory, garlic, and vinegar. Whisk in the olive oil to blend. Season to taste with salt and pepper.
Combine the tomato wedges, cherry tomatoes, and green beans in a bowl. Add the vinaigrette, toss, and season with salt and pepper to taste. Transfer to a shallow serving bowl or platter, sprinkle with the prosciutto, and garnish with the savory sprigs. Serve immediately.