Special equipment: Vegetable juicer, 12-inch non-stick saute pan
Using a vegetable juicer, juice the squash and reserve the pulp. (You should have about 1/2 cup of pulp and 1/2 cup of juice.) Place the squash pulp in a medium mixing bowl and sprinkle with the flour. Using your fingers, gently squeeze the pulp and flour together until just blended. Gently stir in the squash juice and apple juice until just blended.
In a large bowl, whisk the egg whites to a very soft peak. Add the fig syrup and whisk to a medium-firm peak. Gently fold 1/3 of the egg whites into the batter base to lighten it, being careful not to fully incorporate the egg whites. Gently fold the lightened batter back into the egg whites. Again, be careful not to fully incorporate the mixture. (The more gently you treat the egg whites, the higher the batter will rise.)
Heat a 12-inch non-stick saute pan over a medium heat until hot. With a paper towel dipped in the oil, lightly oil the pan. Working in batches, spoon about 2 tablespoons of the batter into the pan, to form as many 2 to 3-inch round cakes as will fit, and cook until just set. Gently turn the cake over and finish cooking, about 1 1/2 to 2 minutes. The cakes will be very soft. Repeat with the remaining batter.
To serve: Shingle the cakes on heated plates. Place a quenelle of creme fraiche against each serving of cakes and drizzle with fig syrup. Sprinkle each plate with Indian maple sugar.