Combine milk and cream in a saucepan and warm over low heat until foam forms around the edges. Remove from heat.
Combine egg yolks, sugar, scraped vanilla bean seeds, and chestnut flour in a large bowl and beat with an electric mixer until frothy. Gradually pour warm milk into egg yolk mixture, whisking constantly.
Return mixture to saucepan; add scraped vanilla bean and cook over medium heat, stirring with a wooden spoon until the mixture gels slightly and coats the back of the spoon. If small egg lumps begin to show, remove from heat immediately.
Pour mixture through a sieve or fine strainer into a bowl. Cover, and chill for several hours or overnight.
Pour mixture into an ice cream maker, and freeze according to manufacturer's instructions. Transfer to a sealed container, and freeze until firm. If the gelato is too firm, place it in the refrigerator until it reaches the desired consistency before serving.