Preheat oven to 350F. Lightly grease a 13-by-9-inch cake pan; line with parchment paper.
Whisk flour, ginger, baking soda and salt in a bowl; set aside.
Using an electric mixer, beat eggs and sugar at medium speed until pale and thick, about 5 minutes.
Heat milk in microwave until hot. Stir in butter. While mixing at low speed, drizzle hot milk into egg mixture; mix until smooth. Stir in vanilla. Sift a third of flour mixture into egg mixture; stir to blend. Repeat twice with remaining flour.
Scrape batter into pan; bake until a cake tester comes out clean, 25 to 30 minutes.
Let cool in pan for 10 minutes, then turn out onto a rack; let cool completely.