In blender container, place 3 cups watermelon and blend until smooth. Press through fine-mesh strainer into bowl; discard pulp. Pour 1 1/2 cups watermelon juice into small saucepan. Add shallots, chili sauce, soy sauce, fish sauce, lime juice, and gingerroot to saucepan. Bring to a boil; reduce heat to medium and cook, uncovered, until liquid is reduced by about one half, 20 to 25 minutes. Stir in sesame oil. Remove from heat and let cool.