Preheat oven to 350°F (180°C). Line 2 baking sheets with parchment paper.

In the bowl of a stand mixer fitted with the paddle attachment, beat melted butter, granulated sugar, egg, brown sugar, oil, vanilla, and salt at medium speed until well combined, 1 to 2 minutes, stopping to scrape sides of bowl.

In a medium bowl, whisk together flour, baking powder, ground ginger, and baking soda. Add flour mixture to butter mixture all at once; beat at low speed just until combined. Fold in chopped walnuts and candied ginger. Using a 1½-tablespoon spring-loaded scoop, scoop dough, and roll into balls (about 25 grams each); place at least 1 inch apart on prepared pans. Flatten into ½-inch-thick, 2-inch-wide rounds; press a walnut half into center of each round. Brush tops with egg white; sprinkle with sesame seeds.

Bake until golden brown, 10 to 12 minutes. Remove from pans, and let cool completely on a wire rack.