Boil pasta. Bring a large pot of salted water to a boil and cook pasta al dente. Reserve 1/2 cup of pasta water before straining.
Prep the sauce. Combine eggs, Parmesan cheese, a pinch of salt, and pepper, and optional cream in a medium-sized bowl. Whisk thoroughly and set aside.
Fry bacon. Meanwhile, cook bacon in a large stainless pan or cast iron skillet on medium heat until fully cooked, remove from the skillet, and place on a paper towel-lined plate to drain. Keep 1-2 tablespoons of bacon grease in the skillet and discard the rest.
Cook chicken. Add sliced chicken. Season with salt and pepper and cook until chicken is fully cooked through (about 5-7 minutes). The bacon fat should brown the chicken. in the last 1-2 minutes, add garlic and a pinch of olive oil. Cook until fragrant.
Assemble and serve. Turn the heat down to low and let the pan cool for 2-3 minutes. Remove from the heat if necessary. Return the bacon to the pan and add the warm linguine. Add the egg mixture to the pan along with 1/4 cup of pasta water. Toss with the pasta until fully mixed and creamy. Add more pasta water if sauce feels too thick. Top with fresh basil and serve immediately with extra Parmesan cheese.