Season turkey and tomatoes with salt and pepper. Grill turkey on the rack of an uncovered grill directly over medium heat for 12 to 15 minutes or until an instant-read thermometer inserted near the center reads 170 degrees F, brushing with pesto the last 5 minutes of grilling.
Meanwhile, toss tomatoes with olive oil in a large bowl. Grill, cut side down, over medium heat about 5 minutes per side until slightly charred. Add lemon, cut side down, for the last 5 minutes of grilling.
Slice turkey; arrange on platter with tomatoes, lemons, and avocado slices. Squeeze lemon halves over avocado and tomatoes. Makes 6 servings.