Cut the chicken breasts into bite-size (1-inch) pieces. Set aside.
Place the green beans in a large skillet and add just enough water to come up the sides but not cover them. Bring the water to a boil over HIGH heat, reduce the heat to LOW and cover the skillet. Simmer for 3 to 4 minutes, or until the green beans are just barely fork tender but still crisp (they will finish cooking later). Transfer the green beans to a colander to drain.
In the dry, empty skillet, heat 2 tablespoons of olive oil over MEDIUM-HIGH heat. Add the chicken and season it with salt and pepper. Cook, stirring occasionally, for 5 to 6 minutes, or until lightly golden and cooked through. Remove the skillet from the heat and transfer the chicken to a plate to rest.
Add the remaining oil to the skillet and return it to MEDIUM heat. Add the partially cooked green beans and cook, stirring, for a minute or two. Add the tomatoes and cook, stirring frequently, just until the green beans are crisp tender and the tomatoes are beginning to blister, about 3 to 4 minutes. Remove the skillet from the heat.
Return the chicken and the juices from the plate to the skillet and toss with the veggies. Add the pesto (off the heat) and stir again to coat everything with the pesto.
Sprinkle with the Parmesan and serve.