Blanch the Broccoli Rabe: Bring a large pot of salted water to a rolling boil. Add the broccoli rabe stems first and cook them for about 30 seconds. Then, add the leaves and cook for another 30 seconds. Use a slotted spoon or tongs to remove the stems and leaves, and transfer them to a colander to drain thoroughly.
Cook the Pasta: Bring the same pot of water back to a boil and add the orecchiette. Cook the pasta for about 2 minutes less than the package instructions recommend, as it will finish cooking in the sauce. Reserve ½ cup of the pasta water, then drain the pasta.
Toast the Almonds: Heat a large skillet over medium heat. Add the sliced almonds and toast them, stirring occasionally, until they are golden and fragrant, which should take about 5 minutes. Remove the almonds from the skillet and set them aside in a small bowl.
Cook the Onions and Garlic: Return the skillet to the stove and increase the heat to medium-high. Add 1 tablespoon of olive oil, followed by the sliced red onion. Cook the onion, stirring occasionally, until it is soft and caramelized, which takes about 10 minutes. Stir in the garlic and red chili flakes and cook for another minute, until the garlic is lightly golden and aromatic.
Combine the Ingredients: Add the remaining 3 tablespoons of olive oil and the blanched broccoli rabe to the skillet. Stir everything together and cook for about 1 minute to heat through. Add the drained pasta and ¼ cup of the reserved pasta cooking water. Toss the mixture well to coat the pasta in the sauce. Scrape up any browned bits from the skillet to enhance the flavor. Continue tossing until the pasta is heated and the sauce clings to it evenly, about 1 minute.
Serve the Dish: Transfer the pasta to serving plates or bowls. Top with the toasted almonds, a generous grating of Parmigiano-Reggiano, a squeeze of fresh lemon juice, and a drizzle of extra olive oil. Serve immediately for best results.