Step 1: Cook the pasta. Bring a large pot of generously salted water to a rolling boil. Add linguine or fettuccine and cook according to package directions until al dente. Reserve about 1 cup of the starchy pasta water before draining the noodles. Set aside.
Step 2: Prepare and cook the chicken. While the pasta cooks, pat the chicken breasts dry with paper towels. Season both sides generously with salt, black pepper, and a pinch of red pepper flakes if desired. Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Add the chicken and cook for 5-7 minutes per side, or until golden brown and cooked through (internal temperature reaches 165°F / 74°C). Remove from skillet, tent loosely with foil, and let rest for 5 minutes. Slice or shred into bite-sized pieces.
Step 3: Blend the creamy avocado-lime sauce. In a high-speed blender or food processor, combine the scooped avocado flesh, fresh lime juice, chopped cilantro, minced garlic, chicken broth, and heavy cream. Blend until the sauce is completely smooth, vibrant green, and creamy. Taste and adjust seasoning with more salt, pepper, or lime juice if needed. If the sauce is too thick, add a tablespoon or two of the reserved pasta water until it reaches your desired consistency.
Step 4: Combine and serve. Return the drained, hot pasta to the large pot or a spacious serving bowl. Pour the freshly blended avocado-lime sauce over the pasta and toss gently but thoroughly to coat every noodle. Add the sliced chicken to the pot and toss again to combine. If the sauce seems too thick, add more reserved pasta water, one tablespoon at a time, until it coats the noodles beautifully. Serve immediately, garnished with extra fresh cilantro, a sprinkle of red pepper flakes, and fresh lime wedges on the side.