Place the peach pieces in the freezer until ready to use.
Combine cream, half-and-half, sugar, cardamom pods, lavender, and salt in a saucepan and heat on medium-low until just simmering, barely boiling.
Stir the mixture occasionally. Once lightly boiling, take the pan off the heat and let the flavors steep together while it cools, for about half an hour.
Strain the cream to remove the lavender flowers and cardamom pods. Add the strained cream back to the pot and heat over medium, until barely steaming.
Crack the eggs into a separate bowl and beat until smooth. Next, temper the eggs by slowly drizzling the hot cream into them with one hand, one ladle at a time, constantly whisking with your other hand. Its important to combine them slowly and keep whisking to avoid scrambling the eggs.
Once a few ladles of cream have been added to the eggs, add the egg/cream mixture back to the sauce pan.
Cook the mixture over medium-low heat, constantly whisking, until it begins to thicken. It is done when it reaches 170 °F, and can thickly coat the back of a spoon. Make sure this process is not rushed, as letting the mixture boil will scramble the eggs.
Once finished cooking, let the mixture cool to room temperature. Then, transfer to a bowl and chill in the fridge, at least 2 hours.
Finally, freeze and churn the mixture in an ice cream machine, which usually takes 15-20 minutes. When the ice cream is 75% frozen, add in the frozen peach pieces and let them combine with the custard.
When the custard is finished freezing, scoop into a container and place in the freezer until ready to make the sandwiches.