Start by heating olive oil in a large heavy skillet over medium heat. Add the sliced garlic and cook it slowly while stirring until it turns lightly golden and fragrant. Remove the garlic with a spoon and place it on paper towels while leaving the flavorful oil in the skillet.
Season the cubed chicken with salt and black pepper. Add the chicken to the hot garlic-infused oil and cook it while stirring occasionally until it becomes golden and fully cooked.
Once the chicken is cooked, stir in the thawed broccoli and red pepper flakes. Let everything cook together for a few minutes until the broccoli is heated through and coated with the flavorful oil.
Meanwhile, bring a large pot of salted water to a boil. Add the spaghetti and cook according to the package directions until it reaches an al dente texture.
Drain the cooked pasta and add it directly to the skillet with the chicken and broccoli mixture. Toss everything together so the spaghetti absorbs the garlicky oil and flavors.
Finally, sprinkle the crispy garlic slices back on top of the pasta. Taste and adjust seasoning with salt and pepper before serving.