Bring a large pot of salted water to a boil.

In a large skillet, heat oil over medium-high heat; add onion, reduce heat to low and cook, stirring occasionally, until tender and golden, about 20 minutes. Meanwhile, cook cauliflower in the boiling water until tender, about 6 minutes. Using a mesh strainer or slotted spoon, transfer cauliflower to a bowl. Add broccoli to the boiling water and cook until just tender yet still firm to the bite, about 4 minutes; using slotted spoon, transfer broccoli to a second bowl. Reserve pot of water.

When onion is tender and golden, add cauliflower, cinnamon, cloves and generous pinch pepper; cook over medium heat, breaking apart cauliflower florets into small pieces, 2 minutes. Remove pan from heat.

Return pot of water to a boil. Add pasta and cook until pasta is about 2 minutes from al dente, then add broccoli to pot with pasta; continue to cook until pasta is al dente. Reserving 1/2 cup pasta cooking liquid, drain pasta and broccoli; transfer to a large serving bowl. Add cauliflower mixture, scraping all of the oil from pan into bowl, then add 1/4 cup pasta cooking liquid, cheese and egg yolks; quickly toss to combine well. Add more pasta cooking liquid to moisten, if desired. Season with salt and pepper to taste. Serve immediately.