Preheat oven to 350 degrees F. Spray 9-inch round nonstick cake pan with nonstick cooking spray. Sprinkle brown sugar over bottom, coating evenly.
Peel, core, and halve pears. Slice pears crossswise into 1/3-inch-thick slices. Evenly spread pears in pan, covering bottom of pan.
Grate teaspoon (or more, to liking) of fresh ginger and sprinkle evenly over pears, letting juice drip onto pears, as well.
In medium bowl, mix together flour, ground ginger, baking powder, baking soda, and salt. Set aside.
In large bowl, using electric mixter, beat together oil, sugar, lime zest, and coconut extract. Beat in whole egg and egg white until thick.
Now, turn off mixer and using a spatula, alternately fold flour mixture and buttermilk into egg mixture, until just blended.
Pour batter over pears, smoothing top to cover pears completely. For more fruit with every slice, leave out about 3/4 cup of the batter. Bake until a toothpick inserted in center comes out nearly clean, about 30-35 minutes. Transfer to a wire rack. Let cake cool in pan on rack for 8-10 minutes, then carefully run a knife around cake to loosen. Invert cake onto a platter. Let cool slightly before slicing.