Prepare the ice cream: Place ice cream in a microwave-safe glass bowl or measuring cup. Microwave on HIGH 10 seconds. Let stand 10 seconds. Microwave on HIGH another 10 seconds. Working quickly, add matcha, and whisk until completely combined and dissolved. Cover with plastic wrap, placing plastic wrap directly onto ice-cream surface. Freeze 3 hours or overnight.
Prepare the pâte à choux: Preheat oven to 400°F. Sift together flour and ginger into a medium bowl. Bring water, butter, and salt to a boil in a medium saucepan over medium. Reduce heat to low, and stir in flour mixture all at once. Cook, stirring constantly with a wooden spoon, until mixture is smooth and pulls away from sides of pan, 30 seconds to 1 minute. Remove from heat; cool 5 minutes.
Transfer mixture to bowl of a heavy-duty electric stand mixer fitted with paddle attachment. Beat on medium-low speed until bowl is no longer hot to the touch, about 5 minutes. Add eggs, 1 at a time, beating on medium speed until smooth after each addition. Spoon mixture into a piping bag fitted with a 1/2-inch round tip. Pipe 24 (1 1/2-inch-round) mounds onto a baking sheet lined with parchment paper. Bake at 400°F for 15 minutes. Reduce oven temperature to 325°F; bake until golden brown and set, about 20 minutes. Cool completely on a wire rack, about 10 minutes.
Prepare the fudge sauce: Stir together cream, corn syrup, sugar, cocoa, salt, and half of chopped bittersweet chocolate in a saucepan. Bring to a boil over medium-high, stirring constantly. Reduce heat to medium-low, and simmer, stirring often, 3 minutes. Remove from heat. Stir in butter, vanilla, and remaining chopped bittersweet chocolate. Let stand until thickened, 10 to 15 minutes.
Assemble: Cut each profiterole in half horizontally. Place 3 bottom halves on each of 8 plates. Top each with 2 tablespoons of the ice-cream mixture; cover with top halves of profiteroles. Drizzle each with 2 teaspoons of the fudge sauce. Place powdered sugar in a fine wire-mesh strainer, and sift over profiteroles, if desired.