Bring a large pot of water to a boil for the pasta.
While waiting for the water to boil, melt butter in a small skillet over medium heat. Once melted, add two cloves of crushed garlic and stir occasionally until golden brown (5-8 minutes). Remove the garlic and set the skillet aside.
In a large bowl, mix together the eggs, egg yolks, and cheese until no streaks remain from the eggs. Add in the garlic-infused butter and stir to combine.
Cook the pasta according to the instructions on the package. Reserve ½ cup of pasta water before draining.
Place a bowl over a pot of simmering water on medium-low heat (make sure the bowl can sit on top without touching the water).
Add the sauce, pasta water, and drained pasta to the bowl. Stir constantly and cook for a few minutes until the sauce slightly thickens (the bowl should never touch the boiling water).