In an 8-inch square, heavy aluminum cake pan placed over medium heat, melt 2 Tbs. of the butter. Add the granulated sugar and cook, stirring occasionally, until the sugar melts and turns light brown, 5 to 7 minutes. Arrange the pear slices in the pan in 4 overlapping rows. Set aside.
Preheat an oven to 350°F.
Over a sheet of waxed paper, sift together the flour, baking soda, ground ginger, cardamom, cinnamon and salt. Stir in the fresh ginger. Set aside.
In the bowl of an electric mixer fitted with the flat beater, beat the remaining 6 Tbs. (3/4 stick) butter until creamy. Add the brown sugar and beat until the mixture is pale and fluffy. Slowly drizzle in the egg, beating each addition until incorporated before continuing. Beat in the molasses.
Reduce the speed to medium-low and add the flour mixture in 3 additions, alternating with the milk and beginning and ending with the flour. Beat just until combined. Pour the batter on top of the pears and spread it evenly to the edge of the pan. Bake until the top of the cake is puffed, 35 to 40 minutes. Transfer the pan to a wire rack and let cool for 10 minutes.
Run a table knife around the edge of the pan and shake it to make sure the cake is not sticking. (If it is, set the pan over low heat and heat for 1 to 2 minutes, gently shaking it until the cake is free.) Place a serving plate upside down on the pan. Wearing oven mitts, invert the plate and pan together. Lift off the pan. Dislodge any pear slices that stick to the pan and arrange them on top of the cake. Serve at room temperature.
Sprinkle with confectioner's sugar on top and around cake on plate with a few sprigs of mint for color.