Bring a large pot of salted water to a boil and cook the pasta according to package directions until al dente; drain and reserve.
While the pasta cooks, melt the butter in a sauté pan over medium heat.
Add the finely chopped shallot and sauté about 2 minutes until 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.
Whisking constantly, slowly pour in the milk. Season with salt and pepper and cook until the mixture begins to thicken.
Reduce the heat slightly, stir in the heavy cream, drained roasted red peppers, and the 1/2 cup shredded Parmesan; cook just until the Parmesan melts.
Carefully transfer the sauce to a blender and blend until the peppers are nearly pureed, leaving a few small chunks if desired. Return the sauce to the pot.
Add the drained pasta to the sauce and toss to evenly coat; cook over low heat 1 minute to allow the sauce to soak into the pasta.
Toss in 1 tablespoon of the chopped basil and parsley.
Plate the pasta, top with the grilled chicken strips, sprinkle with additional Parmesan and the remaining chopped basil and parsley, and serve immediately.
To grill the chicken: preheat a grill to 425°F. Pound thicker parts to even thickness or slice thick breasts in half. Brush with olive oil and season with salt and pepper.
Oil the grill grates and grill the chicken, turning once halfway through, until an instant-read thermometer registers 165°F, about 7–8 minutes total. Let rest 5–10 minutes before slicing into strips.