Fill a saucepan two-thirds full with water and add 2 teaspoons salt; bring to a boil.
Add beans to boiling water and boil until beans are tender to the bite, 4 to 8 minutes. Use tongs to lift beans out of water and spread them out on the towels to let them cool. Add snap peas to boiling water and cook until they're bright green and just slightly softened, about 2 minutes. Drain and let cool.
Season the tomatoes and mozzarella with a pinch or two of salt. Put them in a large mixing bowl and add the beans, snap peas and ½ teaspoon salt. In a small bowl, whisk together pesto, lemon juice and olive oil. Pour over vegetables and mix well. Season with several grinds of fresh pepper. Toss well and serve.