Step 1In a non-reactive bowl, whisk together the oil, vinegar, mustard, salt and several grinds of black pepper. Stir in the onion, basil, capers and garlic. Gently fold in the tomatoes. Cover and refrigerate at least 3 hours before serving, up to overnight.
Step 1Halve the avocados and discard the seeds. Brush the cut interior of the avocados with a layer of mayonnaise (this will also keep them from browning before grilling). Season each half with a sprinkling of salt and a couple grinds of pepper.
Step 2Heat a grill over medium-high heat until hot.
Step 3Place the avocados, cut-side down on a grill rack. Grill the halves for a couple of minutes. Using tongs, rotate the halves slightly for good grill marks and grill for another couple of minutes. Remove to a serving platter, cut-side up.
Step 4Spoon the tomato salad over the avocados and serve immediately.