In a large pot of salted, boiling water, cook the pasta to al dente, according to the package directions.
Drain the water, then rinse the cooked pasta under cold water to cool it off completely.
Drain the water and return the cooked pasta to the pot.
In a large, shallow bowl, add the chicken breasts and season both sides with 1/2 teaspoon of the salt, 1/4 teaspoon of the black pepper, and 1 tablespoon of the pesto.
Let the seasoned chicken breasts stand, about 5 minutes.
In a large skillet over medium-high heat, add the vegetable oil until it is sizzling hot, about 2 minutes.
Add the seasoned chicken breasts to the heated vegetable oil and sear until they reach an internal temperature of 165 degrees F, their juices run clear, and are golden-brown, about 5 minutes per side.
Transfer the cooked chicken breasts to a cutting board and cut them into strips.
In the same large skillet, melt the butter until it starts to bubble, about 1 minute.
Add the onions, the garlic, and the tomatoes to the melted butter and sauté until the tomatoes burst, about 4-5 minutes.
Add the heavy cream, the chicken broth, and the remaining pesto sauce to the onion mixture and whisk to combine.
Bring the sauce to a simmer and cook, stirring occasionally, until it has thickened to the desired consistency, about 3-4 minutes.
Taste the sauce and season with the remaining salt and the remaining black pepper, as needed. Stir to combine.
Add the cooked chicken strips to the sauce and toss well to coat. Cook, stirring constantly, until it is heated through, about 1 minute.
Add the cooked pasta to the chicken mixture and toss to combine. Cook, stirring constantly, until it is heated through, about 1-2 minutes.
Serve the chicken mixture garnished with the parmesan and the parsley.