Start by making the vinaigrette: In a bowl, whisk the olive oil, lemon zest and juice, mustard and shallot; season to taste with salt and pepper.
For the salad, combine the cherry tomatoes with salt and pepper to taste in a small bowl. Set aside.
Preheat a grill to high. Fill a large saucepan with 1 1/2 quarts water; add 1 tablespoon salt and the vinegar. Bring to a simmer over medium-high heat. Add the farro and gently simmer until al dente, 15 to 20 minutes; drain well. While the farro is warm, immediately mix with about three-quarters of the vinaigrette (set aside the rest to punch up or remoisten the salad later if necessary).
While the farro is cooking, brush the corn with the vegetable oil. Season with salt and pepper. Cook on the hot grill, turning with tongs, until charred, 10 minutes. Cut off the kernels.
Stir the corn kernels, tomatoes, olives, pistachios and chives into the farro. Gently toss to combine; add more vinaigrette and season with salt and pepper. Garnish with the basil.