Beat eggs and egg yolks in a medium sized bowl. Next add sugar to the bowl and whisk together.
Add heavy cream and half and half to a large saucepan and heat gently; do not let come to a boil.
Temper eggs by slowly pour 1 cup of hot heavy cream mixture into egg mixture while whisking quickly. This will raise the temperature of the eggs carefully without scrambling them. Then while stirring, pour the tempered egg mixture slowly into the remaining heavy cream mixture in the saucepan.
Gently heat ice cream base over medium low heat, stirring constantly (especially the corners of the pot), until thickened and reaches 170F on a digital thermometer. The mixture should look silky and coat the back of a spoon so when you run your finger through it, it doesn’t come back together.
Remove from heat and add small pinch of flour and vanilla.
Strain through a fine-mesh sieve (to catch any cooked egg pieces) and chill in an ice bath (a bowl set inside of a bowl of ice water) until it’s cooled to room temperature. Then cover with plastic wrap and move into the refrigerator until completely cold, about 2-3 hours.
Once completely cooled, add your mixture to an ice cream maker according to the manufacturer's instructions and freeze.