Set a pot of heavily salted water to boil for the pasta. Meanwhile, to a large heatproof mixing bowl, add the egg yolks, heavy cream, 3 Tbsp. Grana Padano, and a generous pinch each of salt and black pepper; whisk to combine. Set the bowl aside and let come to room temperature.
Set a paper-towel-lined plate next to the stove. In a large skillet over medium heat, add the chopped bacon and cook, stirring occasionally, until some of the fat has rendered and the bacon is crisp, about 13-14 minutes. Remove the bacon with a slotted spoon or spatula and transfer to the paper-towel-lined plate (reserve about 2 Tbsp. fat in the skillet). Lower the skillet to medium heat and add the broccoli; season lightly with salt. Cook, stirring occasionally, until the broccoli is slightly softened but still bright green, about 3-4 minutes. Stir in the garlic and cook until fragrant, one minute more. Turn off the heat.
Season the boiling pasta water with salt, add the spaghetti and cook just until al dente. Using tongs, transfer the spaghetti in batches to the bowl with the egg mixture, stirring rapidly with the tongs after each addition. Do not drain the pasta water.
Add the broccoli-garlic mixture and the crisp bacon to the pasta and toss well. Adjust the seasoning and loosen the sauce as needed using one tablespoon at a time of the pasta cooking water. Divide among two plates and garnish with more cheese and black pepper as desired.