For the tangerine curd: In a small saucepan, bring 1 inch of water to a simmer over medium heat. In a medium bowl, whisk the 3⁄4 cup of sugar with the egg yolks and a pinch of salt until smooth. Stir in the lemon and tangerine juices and then place the bowl over the saucepan. Cook, stirring occasionally, until the curd is thickened and an instant-read thermometer inserted into the curd reads 180°, about 16 minutes.
Remove the bowl from the saucepan and stir in the butter, one cube at a time, allowing each cube to melt before adding another. Whisk in the olive oil and then place a sheet of plastic wrap on the surface of the curd. Refrigerate until chilled, at least 4 hours.
Meanwhile, make the cola syrup: In a small saucepan, stir the remaining 1⁄3 cup sugar with the cola, bitters, and ground and stick cinnamon. Bring to a simmer over medium heat and then cook, stirring, until slightly reduced and thickened, about 5 minutes. Remove the pan from the heat and let the syrup cool to room temperature. Pour the syrup through a fine sieve into a small bowl and refrigerate until ready to use.
To serve, process ice in a blender until very finely crushed. For each serving, spoon about 3 tablespoons of the tangerine curd into the bottom of a glass and top with 1 cup of shaved ice. Drizzle with about 2 tablespoons of the cola syrup and serve immediately.