Bring a large pot of salted water to a boil and cook the penne according to package directions; drain and set aside.
Heat the olive oil in a very large skillet over medium-high heat.
Add the diced chicken and cook, stirring and flipping intermittently, until cooked through, about 5 minutes.
Add the diced red bell pepper and prepared green beans; cover and steam until the vegetables begin to soften, about 4 minutes.
Uncover, add the minced garlic and cook, stirring nearly continuously, until fragrant, about 1 minute.
Season with kosher salt and freshly ground black pepper, then stir to combine.
Add the cooked penne to the skillet and stir to combine with the chicken and vegetables.
Sprinkle the shredded asiago evenly over the pasta mixture and toss/stir nearly continuously until the cheese melts and coats the pasta.
Stir in the minced fresh basil, taste, and adjust seasoning with more salt or pepper if desired; serve immediately.
