Bring half n half and coconut milk with the cinnamon sticks just to a boil in a saucepan, then remove from heat and let steep, uncovered, 30 minutes.
While the cinnamon steeps, simmer mangos, vodka, and 1/2 cup sugar in a 1 quart sauce pan, partially covered, until mangos are tender, about 15 minutes. Cool slightly, then puree mixture in a blender, scraping down sides occasionally (use caution when blending hot liquids), until smooth. Whisk together yolks and remaining sugar in a bowl, then add half-and-half and coconut mixture( do not strain) in a stream, whisking.
Pour custard into 3 quart saucepan and cook over moderately low heat, stirring with a wooden spoon, until it registers 170F on thermometer, about 8 minutes, immediately pour custard through a fine-mesh sieve into a metal bowl and whisk in mango puree with vanilla, ginger, paprika, and chili pepper. Chill with a plastic wrap cover, stirring occasionally, for approximately 1 1/2 hours. This step is important since you do not want ice crystals to form in the ice cream during preparation, so this means that the mixture should be completely cooled.
Freeze custard in ice cream maker (in batches if necessary), then transfer to an airtight container and put in freezer harden, about 12 hours. Do not open containers during this process to prevent ice crystals from forming.