In a saucepan, heat up butter and olive oil on medium-low heat until melted.
Add cream, garlic, and white pepper to the saucepan and bring it to just under a boil.
Reduce the heat and let it simmer for about 5 minutes, stirring often, until the mixture is slightly reduced.
Stir in Parmesan cheese into the cream mixture and let it simmer for 8 to 10 minutes, until the sauce is thickened and smooth.
Add mozzarella cheese to the sauce and cook and stir for approximately 5 minutes, until the cheese is melted.
In a large pot, bring lightly salted water to a boil.
Add angel hair to the boiling water and cook while stirring occasionally until it's cooked through but still firm to the bite, for around 3 to 5 minutes.
Drain the pasta and transfer it to serving plates.