Remove thawed grated ube from package and place in a microwave-safe bowl. Add 2 tablespoons water.
Microwave until cooked, 3 to 4 minutes. The mixture should feel smooth between your fingers, like thick mashed potato.
Measure out 1¼ cups cooked ube, and reserve the remainder for another use.
Transfer the 1¼ cups cooked ube to a food processor. Add the granulated sugar and light brown sugar and process on low until smooth.
Add the eggs, melted butter, heavy cream, ube flavoring, and salt, and process just until combined and no streaks remain. Filling will be liquidy, but trust the process!
Pour filling into prepared crust.
Bake pie on center rack until slightly puffed around the edges and just set in the center, 30 to 35 minutes. Cool completely before serving at room temperature or chilled. Covered and refrigerated, the pie will keep up to 2 days.