Butter and flour six ramekins or muffin cups (1-cup size). In a 5- to 6-quart pan over high heat, melt 1/2 cup butter; stir often until butter is golden brown with dark brown flecks, 3 to 5 minutes.
Remove at once from heat and let stand until just warm, 15 to 20 minutes.
Meanwhile, place hazelnuts in an 8- or 9-inch-wide pan and bake in a 350 regular or convection oven until golden beneath skins, about 10 minutes.
Pour onto a clean linen towel and let stand until cool enough to handle, then rub nuts in towel to remove loose skins.
In a food processor, whirl nuts with flour, 1/2 cup granulated sugar, lemon peel, salt, and cinnamon until finely ground.
In a large bowl, with a mixer on high speed, whip egg whites until frothy. Gradually add remaining 1/4 cup granulated sugar and continue to whip until mixture holds soft peaks. Gently stir in nut mixture. Gradually add brown butter, stirring until well blended. Spoon batter equally into ramekins. Overlap fig slices on tops in a circular pattern.
Bake in a 350 regular or convection oven until cakes are lightly browned and a toothpick inserted in the center comes out clean, about 30 minutes (25 minutes in convection oven). Set cakes on a rack to cool for 10 to 15 minutes.
To remove cakes from ramekins, run a thin-bladed knife between each cake and ramekin to loosen; invert to unmold, then set upright.
