Slice and core peppers. Rough chop the onion. Add peppers, onion, almonds, and 1/4 cup of olive oil to a blender or small food processor. Blend for 1 minute until smooth. Slowly add more olive oil if needed. You are looking for the consistency of smoothie you can drink with a straw. Set aside.
Cook your pasta 2 minutes less than what al dente says on the package. Reserve 2 cups of pasta water. Strain and rinse with cool water to stop the cooking. Set aside.
Season chicken breast with cajun seasoning. Heat large non-stick stock pot over medium-high heat. Add 1 Tb EVOO to the pan. When heated, add the chicken. If you chicken breast is more than 1" thick, butterfly it before seasoning and cooking.
Cook for 4 - 5 minutes on one side. The chicken will start to turn white half way up the side. Flip and repeat. It should take 8 - 10 minutes to cook your chicken to 165°. Use a thermometer if you are unsure. Remove from pan and rest on a plate.
Slice your garlic. Add a teaspoon of EVOO to the same pan. Cook for 90 seconds.
Add 1 cup of pesto sauce to the pan along with 1 ladle of paster water. Scrap the bottom of the pan to incorporate the fond that developed when cooking the chicken. Continue to cook the sauce down to its original consistency. Repeat this process 2 more times in order to develop the flavors in the sauce.
Add your pasta back to the pot. Add more water if needed. Stir and cook pasta to al dente. Add half of your cheese to the pot and incorporate. Taste the pasta and add more salt if needed.
Slice your chicken against the grain. Serve your pasta and top with chicken and the remaining cheese.