Heat 2 teaspoons of the oil in a 12- to 14-inch nonstick skillet over medium heat, and then add the asparagus and sauté until crisp tender, 3-5 minutes depending on thickness. Season with a little salt and pepper and remove to a plate.
Drizzle 2 more teaspoons of oil into the pan, add the mushrooms and sauté until they release their liquid and are golden brown, about 5-7 minutes. Add a light sprinkle salt and pepper, and then transfer to the plate with the asparagus. (Tip: letting the mushrooms sit, stirring only occasionally, allows them to better caramelize and become golden brown.)
Add the final 2 teaspoons of oil to the pan, followed by the chopped chicken, seasoning again with a light sprinkle of salt and pepper. Cook for approximately 5 minutes, flipping a few times, until the chicken is just cooked through.
Stir in the tomatoes and return the cooked asparagus and mushrooms to the pan. Add the pesto and stir to evenly coat. Cook 1-2 minutes more or until everything is heated through.