Arrange the sliced tomatoes, slightly overlapping, around the perimeter of a serving platter or individual serving plates; sprinkle lightly with coarse salt; set aside.
In a medium saucepan over medium-high heat, bring water to a boil. Add the green beans and cook for 5 to 6 minutes (depending on the degree of doneness you want). Remove from heat. Drain the beans in a colander; drain.
Rub the inside of a large bowl with the cut side of the garlic; reserve for other use. Add the warm beans and 2 tablespoons of chopped mint; toss to coat.
Using the coarse shredding side on a hand grater, grate the hard-cooked eggs into a separate bowl. Add the olive oil, the remaining 4 tablespoons of chopped mint, salt, and pepper to taste. Stir to blend well. Add to the beans; stir to coat evenly.
Mound the beans in the center of the serving platter. Garnish with a few whole mint leaves and serve immediately.
Makes 6 to 8 servings.