Step 1

Preheat oven to 325°. Toast cashews on a rimmed baking sheet, tossing halfway through, until golden brown, 12–14 minutes.

Step 2

Meanwhile, toast allspice and 2 cinnamon sticks in a medium saucepan over medium heat, stirring often, until fragrant, about 3 minutes. Add cashews, milk, sugar, and kosher salt and reduce heat to low. Cook, stirring occasionally, until cashews soften and flavors infuse (do not let milk boil), 20–25 minutes. Let cool slightly.

Step 3

Transfer mixture to a blender and purée on high speed until very smooth and the consistency of heavy cream. Wipe out saucepan. Strain chilate through a fine-mesh sieve back into saucepan. Reheat chilate over medium-low, stirring occasionally, until hot.

Step 4

Ladle chilate into mugs and finely grated remaining cinnamon stick over; sprinkle with sea salt.

Step 5

Do ahead: Chilate can be made 1 week ahead. Let cool. Transfer to an airtight container; cover and chill. Reheat over medium-low heat before serving.