Fill a large pot with salted water and bring it to a boil. Cook the pasta following the package instructions until just tender but firm. Save half a cup of the cooking water, then drain the pasta.
While the pasta cooks, heat a tablespoon of olive oil in a large skillet over medium-high heat. Season the chicken pieces with salt and pepper, then cook in the skillet until browned and no longer pink inside, about 6 to 8 minutes. Remove and set aside.
Add a little more olive oil to the skillet if necessary. Toss in the sliced red bell pepper, zucchini, broccoli florets, and snap peas. Stir-fry these vegetables for roughly 5 to 6 minutes until they soften but still have a bit of crunch.
Mix in the minced garlic, dried oregano, crushed red pepper flakes, and cherry tomatoes, cooking everything together for another 2 minutes to meld the flavors.
Turn the heat down to medium-low. Shift the vegetables to one side of the pan. Add butter and olive oil into the cleared space, letting the butter melt completely.
Return the cooked pasta and chicken to the skillet. Add the grated parmesan, lemon juice, and a splash of the reserved pasta water. Stir everything until the sauce coats all ingredients smoothly. If the dish seems dry, add more pasta water as needed.
Dish up the pasta primavera with extra parmesan sprinkled on top, fresh parsley or basil for garnish, and an optional squeeze of lemon juice for added brightness.
