Start by heating the water in order to cook the pasta, as this step takes the longest. Salt the water, if desired.
Meanwhile, in a saucepan over medium heat, melt the butter with the minced garlic. Cook for 3 minutes.
Add the heavy cream and the egg yolk to the saucepan, and lower the heat to medium-low, whisking lightly to ensure the yolk mixes with the cream. Cook for 3 additional minutes. The mixture should be hot, but should not be boiling.
Lower the heat to low.
When the water is boiling, add the fettuccine and cook until the desired texture is achieved.
Once the fettuccine is cooked, add the pasta to the sauce and toss lightly until the fettuccine are fully covered. Add the grated parmesan and continue to toss. Serve while warm.