Preheat the oven to 400 degrees. Peel the garlic cloves. Small cloves can remain whole, and any giant cloves should be cut in half. Roast the tomatoes with the garlic cloves, crushed red pepper, salt and a solid coating of olive oil, around 2 tablespoons, covered with foil. A cast iron or other oven-proof skillet works best here, but a metal sheet pan will work in a pinch. Don't use a glass pyrex - the tomatoes and garlic will not caramelize properly.
As soon as the tomatoes go in, start the pasta water to boil in a large pot, salting well. While waiting for the water to boil, de-stem kale and tear into small, bite-sized pieces. Once the water is boiling, check the tomatoes. The goal is for them to be bursting and releasing some juice and the garlic to be getting soft. It might take until the pasta is almost done, but be sure they are at least softening and a few have burst before adding the pasta to the water. Boil the spaghetti until just shy of al dente.
Right before the pasta finishes, pull out the tomatoes and place on the burner next to the pasta, on the stove, but don't turn on the burner. Using the residual heat in the pan, sauté the kale. It should wilt down without having to turn on the stove as the pasta finishes cooking, with an occasional stir. Stick back in the oven if necessary to finish wilting.
When the pasta is finished, remove and reserve 2 cups of pasta water. Drain the pasta and leave a little water in the bottom of the pot, just enough to cover the bottom, but not so much that the pasta is swimming.
Add the tomatoes and kale to the pasta and stir well. If there isn’t some amount of liquid still coating the bottom, add about half a cup of pasta water now. Pile the cheese in the middle of the top of the spaghetti, and make a small well in the center. This will be the nest for the egg yolk, buying a bit of time to avoid scrambled eggs. Gently separate the yolk from the white, and place the egg yolk on top of the cheese nest. With pasta water close by, and a good pair of tongs, mix quickly and thoroughly — the goal is to get the cheese and yolk evenly incorporated. It’s not as hard as it seems, just continue mixing. Drizzle pasta water in slowly, about 1 tablespoon at a time, and mix with the tongs for about 1 - 2 minutes, until a glossy sauce forms. When the sauce is done should be a little thin, but should have a consistent texture and seem cohesive. Then, by the time it’s at the table and served, it will set up slightly, to be a perfect creamy sauce.