Cook spaghetti according to package directions. Drain and rinse with cold water; set aside.
Slice bell peppers into thin strips. Heat a skillet over medium-high heat. Add 2 tbsp oil, then add peppers and cook until tender, 5-6 minutes.
Slice chicken breasts lengthwise to make thin and pound to even thickness. Season with oregano, salt, and pepper.
Once peppers are done, transfer to a blender or food processor and set aside.
Return skillet to heat and add 2 tbsp oil. Add chicken to skillet and cook until golden brown and cooked through, working in batches if necessary. Transfer to plate and set aside.
Transfer half the basil leaves to the blender with the red bell peppers. Thinly slice the remaining basil and set aside.
Add garlic, pine nuts, sun-dried tomatoes, remaining 2 tbsp oil, salt, and red pepper flakes to blender. Blend on high, adding water as necessary, to form a semi-smooth pesto.
Grate Parmesan cheese.
Return skillet to medium-high heat. Add spaghetti and pesto, tossing to coat. Sprinkle with parmesan and stir to combine.
Slice chicken into strips and serve over noodles. Top with remaining basil.