Cook the gluten-free pasta according to package directions until al dente; drain and set aside.
Heat the olive oil in a large skillet over medium-high heat.
Season the chicken tenderloins with kosher salt and black pepper, then add them to the hot skillet; cook 3 minutes per side until golden brown (about 6 minutes total). Remove chicken to a plate — it will finish cooking later.
Add the sliced red, yellow, and green bell peppers to the same skillet and cook, stirring occasionally, until softened, about 5 minutes.
Push the peppers to the side, then add the butter, minced garlic, and sliced shallot; cook until fragrant and the shallot is softened, about 2 minutes.
Return the chicken to the skillet and add crushed red pepper, chicken broth, lemon juice, and lemon zest. Simmer until the chicken is cooked through and the sauce is slightly reduced, about 5 minutes.
Add the drained pasta to the skillet and toss gently to coat in the sauce and combine with the peppers and chicken.
Divide among bowls and garnish with chopped fresh parsley and grated Parmesan if desired.