Bring a large pot of salted water to a boil. Cook the linguine until al dente according to the package directions. Scoop out 1/2 cup (120 ml) of the starchy pasta water and set it aside, then drain the pasta.
Whisk the whole eggs, egg yolks, Parmesan cheese, heavy cream, salt, and black pepper together in a medium bowl. Set this mixture aside.
Place a large skillet over medium heat and add the chopped bacon. Cook until crisp, which takes about 8 minutes. Use a slotted spoon to transfer the bacon to a paper towel-lined plate. Leave about 2 tablespoons of bacon grease in the skillet.
Add the chicken strips to the skillet in an even layer. Cook for 5 to 7 minutes, stirring occasionally, until browned and the internal temperature reaches 165°F (74°C) on a meat thermometer.
Stir the minced garlic into the skillet with the chicken. Add a splash of olive oil if the pan looks too dry. Cook for 1 minute until fragrant.
Remove the skillet from the heat and let it cool for 3 minutes. This step prevents the eggs from scrambling when they hit the pan.
Return the cooked bacon and drained linguine to the skillet. Pour the egg mixture over the pasta, adding a splash of the reserved pasta water. Toss vigorously with tongs until a smooth sauce forms, adding more pasta water as needed.
Garnish the pasta with fresh chopped basil and serve warm.