Bring a large pot of salted water to a boil.
Meanwhile, heat a small nonstick skillet over medium heat. Place a small crumble of feta into the skillet: if it sizzles, the pan is hot enough for the next step.
Crumble the remaining feta into the pan, placing it around the perimeter and leaving room for the egg in the center. Dollop a few small spoonfuls of pesto onto the feta.
At this point, the feta should be melting and bubbling, and maybe the edges are even taking on a bit of color. That means it's time to crack the egg into the center. Season with black pepper and red pepper flakes. (No need to add salt, the feta is plenty salty).
Cover the pan and cook, reducing the heat to medium-low or low if the feta is browning too quickly, until the white is set, the yolk is still runny, and the cheese is crispy. Use a wide, thin spatula to transfer to a plate.
While the egg is cooking, cook the angel hair in the boiling water according to package directions for al dente (likely ~2 minutes). Drain.
Add the pasta to the now-empty skillet. (If it's still very hot, you can keep it off the heat. Otherwise, turn the heat to low). Add a dollop of pesto and toss to coat the pasta (I like using tongs for this).
Transfer the pasta to your favorite pasta bowl and top with the fried egg. Drizzle with olive oil, if desired, then break the yolk so it too can coat the pasta. Dig in!
