Bring 1 quart of water to a boil for a water bath. Set eight 6-ounce custard cups in two 8-inch square pans.
Heat pumpkin, ginger, cinnamon in a medium saucepan over medium heat until puree sputters and flavors intensify, 3 to 4 minutes.
Whisk in brown sugar, then evaporated milk, and bring to a simmer.
Meanwhile, whisk egg and yolks in a medium bowl. Gradually whisk hot pumpkin mixture into eggs, then pour into custard cups. Set pans in oven and carefully pour in enough hot water to come halfway up sides of cups.
Bake until custards are set, about 30 minutes. Remove custard cups from baking dish and cool slightly.
Lay plastic wrap directly over each custard to prevent a skin from forming. Chill until ready to serve.