Take one of the lemons and peel the rind off the a vegetable peeler. Take the remaining 3 lemons and finely grate the zest with a microplane.
Add the finely grated lemon zest and the sugar to a food processor and blend until combined.
Heat 1 cup of the heavy cream, the half and half and the thickly peeled rind in a saucepan over medium low heat. You want it to just bubble at the edges – do NOT scald or cook it! With the mixture is heating, whisk together the eggs and sugar in a large bowl until pale yellow and smooth. Remove the warmed cream from the heat and slowly stream it into the egg yolks and sugar while whisking the entire time. Remove the thick lemon rind. Then pour the mixture back in the saucepan. Heat over medium-low heat, stirring constantly, until it thickens slightly and can coat the back of a spoon. Stir in the lemon and vanilla extracts.
Transfer the mixture to a bowl and let it cool completely. Whisk in the other cup of heavy cream, then cover it with plastic wrap. Refrigerate it until cold, at least 6 to 8 hours or overnight.
Add the cold mixture to your ice cream maker and churn according to the directions. I use the kitchenaid attachment and churn for 20 minutes. After 20 minutes, stir in the crushed cookies. Spoon the gelato into a freezer safe container and freeze for 4 to 6 hours. EAT.