Place a grill basket on the outdoor grill; preheat the grill and the basket to medium. Place the green beans, wax beans, and cherry tomatoes in a large Ziploc bag. Drizzle with 1 tablespoon olive oil. Season with kosher salt and coarse black pepper to taste. Seal the bag and toss to coat. Set aside.
Fill an 8 quart pot ¾ up with lightly salted water. Bring to a boil. Add the linguini and cook uncovered until al dente, about 11 minutes, stirring occasionally. Drain well. In a small bowl, whisk together 2 tablespoons olive oil, balsamic vinegar, garlic, Dijon mustard, and 1 tablespoon kosher salt. (If using regular salt, use just 2 teaspoons.) Drizzle over warm pasta and toss very well to combine. Working in batches as necessary, place the green beans and wax beans in the grill basket and grill, covered, for 6 to 8 minutes, tossing halfway through. Then grill the cherry tomatoes in the basket for 3 minutes, shaking the basket halfway through.
Cut the grilled green and wax beans in half on the diagonal. Toss with seasoned linguini and grilled cherry tomatoes, and serve.
Great warm or at room temperature.