Preheat oven to 350°F. Melt 1/4 cup of the butter in a 10-inch cast-iron skillet over medium. Add brown sugar, and continue cooking over medium, whisking constantly, until sugar melts, 2 to 3 minutes. Remove from heat.
Toss cranberries, orange zest and juice, 1/8 teaspoon of the salt, and 1/4 teaspoon of the ginger in a medium bowl until fully coated. Spoon cranberry mixture, including juices, evenly over brown sugar mixture in warm skillet.
Whisk together flour, baking powder, cinnamon, and remaining 1 teaspoon salt and 1/4 teaspoon ginger in a medium bowl until combined; set aside.
Beat granulated sugar and remaining 1/2 cup butter with a stand mixer fitted with a paddle attachment on medium speed until light and fluffy, 3 to 4 minutes.
Add eggs to butter mixture, one at a time, scraping down sides of bowl as needed. Beat on medium-low speed until fully combined, about 1 minute. With mixer still running, beat in yogurt and vanilla until fully incorporated, about 1 minute.
Add milk and flour mixture alternately in 3 additions, beginning and ending with flour mixture. Beat on medium speed, scraping down sides of bowl as needed, until batter is fully combined and no lumps remain, about 3 minutes.
Spoon batter evenly over cranberry mixture in skillet, and spread evenly.
Place skillet on a large rimmed baking sheet (optional: lined with aluminum foil), and bake in preheated oven until a wooden pick inserted in center comes out clean and top of cake is golden brown, 40 to 45 minutes.
Cool cake in skillet on a wire rack 10 minutes. Place a serving platter over top of warm cake, and carefully flip skillet over, inverting cake onto platter.
Dust cake with powdered sugar, and serve warm (with vanilla ice cream, optional).