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, but do not let it come to a boil.
Temper eggs by slowly pouring 1 cup of hot heavy cream mixture into the egg mixture while whisking quickly. Then, while stirring, pour the tempered egg mixture slowly into the remaining heavy cream mixture in the saucepan.
Gently heat the ice cream base over medium-low heat, stirring constantly (especially the corners of the pot), until thickened and it reaches 170°F 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 the heat and then strain through a fine-mesh sieve.
Stir the vanilla extract, bourbon if using and cooled peach puree into the custard, and chill over an ice bath until cooled to room temperature. Cover with plastic wrap and refrigerate until completely cold, about 2-3 hours.
Once completely cooled, pour the mixture into an ice cream maker and churn according to the manufacturer's instructions. During the last few minutes of churning, add the reserved fresh peach chunks you refrigerated earlier. Freeze until firm before serving.