Toss cauliflower on a large rimmed baking sheet with enough olive oil to coat and sprinkle generously with salt. Roast until tender and golden, about 20 minutes.
Bring a large pot of water to a boil and add 2 tablespoons salt. Add the spaghetti and cook until al dente, about 10 minutes. Scoop out and reserve 1/2 cup of the cooking water, then drain.
While the spaghetti cooks, whisk the eggs in a mixing bowl large enough to hold the spaghetti. Season with a pinch of salt and add half the cheese. Whisk in 1/4 cup of the pasta water.
Heat a small skillet over medium heat. Pour in enough oil to coat the bottom of the pan. Add the garlic, and cook a few minutes until fragrant (don't let the garlic brown). Stir in the chili paste, capers, and cream. Cook another minute or two then add to the egg mixture and whisk together.
Add the drained spaghetti to the egg mixture and toss vigorously with tongs until the pasta is well coated and the sauce thickens a bit. Stir in the cauliflower and parsley.
Serve in bowls sprinkled with the remaining cheese.
If you can't locate Calabrian chili paste, Asian-style chili-garlic sauce or harissa will work, too. Adjust the amount of Calabrian hot peppers you use depending on the brand, as they seem to vary in intensity.