Step 1Place a large pot of water on the stove to boil, adding enough salt for it to taste like seawater. Meanwhile, in a medium skillet, combine 2 tablespoons olive oil, the garlic and chile flakes over medium heat. Cook for about 3 minutes, until the oil starts to sizzle, then reduce the heat to low and shake the pan, stirring constantly. Cook until the garlic just begins to color, about 2 minutes. Add the chicken stock and increase the heat to high. Cook for about 3 minutes, until the mixture is reduced by half. Remove from the heat and set aside.
Step 2Break each of the eggs into a separate small bowl (you will need four bowls). Heat a 10-inch frying pan over high heat until it's hot enough to sizzle when a drop of water is sprinkled in. Add 2 tablespoons oil; swirl it in the pan to heat. Place the eggs in the pan one at a time, making sure they do not touch each other. Add 1 tablespoon water and immediately cover with a lid and reduce the heat to low. Cook 30 seconds to 1 minute, to desired doneness (the yolks should still be runny). Remove the lid and take the pan off the heat, but keep in a warm place until the pasta is ready.
Step 3Add the spaghetti to the boiling water. Cook to almost al dente, about 5 to 6 minutes. Remove from heat; add the rapini to the pot with the spaghetti. Stir briefly.
Step 4Quickly drain the pasta and rapini over a large bowl, reserving some of the pasta water to use if needed to adjust the consistency of the sauce. Pour the garlic-chicken stock mixture over the drained pasta and rapini. Sprinkle in 2 tablespoons bottarga and toss to combine. Adjust the seasoning if necessary, and if the pasta sauce is a little too dry, add a little reserved pasta water (only enough to keep the pasta moving in the pan).
Step 5Divide the pasta and rapini onto four hot plates. Place an egg on top of each serving. Sprinkle an equal amount of the remaining bottarga over each plate and drizzle with the remaining 1 tablespoon olive oil. Serve immediately.