Bring a large pot of salted water to a boil and cook the linguine or fettuccine according to package directions until al dente; drain and set aside, reserving a small cup of pasta water if needed.
While the pasta cooks, melt the butter in a large sauté pan over medium heat.
Add the finely chopped shallot and sauté about 2 minutes until softened and lightly golden, then add the minced garlic and cook 30 seconds more.
Sprinkle in the flour and cook, stirring constantly, for 1 minute to remove the raw flour taste.
Slowly whisk in the milk, season with salt and freshly ground black pepper to taste, and cook until the mixture begins to thicken.
Reduce the heat slightly, stir in the heavy cream, drained roasted red peppers, and shredded Parmesan, and cook just until the cheese melts.
Transfer the sauce to a blender and blend until the peppers are nearly pureed, leaving a few small chunks if desired; alternatively use an immersion blender in the pot.
Return the blended sauce to the pan, add the drained pasta, and toss over low heat until the pasta is evenly coated and heated through, about 1 minute; add a splash of reserved pasta water if the sauce needs loosening.
Stir in 1 tablespoon of the chopped basil and parsley into the pasta.
Plate the pasta and top with the grilled chicken strips, additional shredded or shaved Parmesan, and the remaining chopped basil and parsley; serve immediately.
To grill the chicken: preheat a grill to 425°F (218°C). Pound thicker parts to an even thickness or slice thick breasts in half through the thickness. Brush both sides with olive oil and season with salt and pepper.
Lightly oil the grill grates and grill the chicken, turning once halfway through, until an instant-read thermometer registers 165°F (74°C), about 7–8 minutes total depending on thickness.
Transfer grilled chicken to a plate, tent with foil and let rest 5–10 minutes before slicing into strips to serve atop the pasta.