1. To make the sauce: In a 2-quart saucepan over medium-high heat, combine the dried cane juice and 2/3 cup water. Stir until the dried cane juice is dissolved. Raise the heat to high but don't stir-just swirl the pan over the heat until the bubbling liquid turns from Champagne-colored to an amber-caramel tone. Remove the pan from the heat and immediately pour in the coconut milk (be careful not to let the liquid boil over). Put the pan back over the heat, stirring. Add the salt and cook over medium heat to dissolve all the sugar. When smooth, remove from the heat and transfer to a pouring cup.

2. To make the filling: In a small, heavy-bottomed saucepan over medium-high heat, combine the coconut milk, brown sugar, salt, and cornstarch. Bring to a simmer, whisking until thickened. Remove from the heat and scrape the mixture into a small bowl. Stir in coconut and vanilla. Cover with plastic wrap, pressing the wrap gently down onto the surface of the pudding. Chill for at least 30 minutes.

3. To make the crust: Preheat the oven to 400°F. In a large bowl, mix the flour, sugar, and salt. Using the big holes of a grater, grate the chilled coconut oil into the flour mixture. Toss to mix, then gradually stir in the ice water just until the dough can be pressed together to form a ball. Divide the dough into 6 pieces, then chill for 30 minutes.

4. Roll out each ball into a 6-inch-long oval, then cut it in half crosswise to make two pieces. Holding the dough in your palm, dampen half of the straight edge, then pull the other side around and seal them to make an open cone. Place 2 tablespoons of the filling into the cone, dampen the top, and press the dough together to make a three-sided pastry. Use a fork to seal the edge. Continue until all 12 are formed. Place them on a baking sheet, brush with water, and sprinkle with the cinnamon and dried cane juice. Bake until the bottoms are browned and the edges are crisp, 20 to 25 minutes.

5. Serve warm, drizzled with the sauce.