SPECIAL EQUIPMENT: One 9 by 2-1/2-to-3-inch springform pan, bottom coated with shortening, topped with a parchment round. Do not grease the sides.
Preheat the oven twenty minutes or more before baking, set an oven rack in the lower third of the oven and preheat the oven to 350°F/175°C.
In a small bowl, place 1/4 cup of the sugar for the nuts. In another small bowl, place 2 tablespoons of the sugar for the meringue. In the bowl of a stand mixer, place the remaining 1/4 cup plus 2 tablespoons sugar.
Spread the pecans evenly on a baking sheet and bake for about 7 minutes to enhance their flavor. Stir once or twice to ensure even toasting and avoid overbrowning. Cool completely. In a food processor, pulse the pecans with 1/4 cup sugar and espresso powder, if using, in long bursts until very fine. Stop before the pecans start becoming oily or pasty. Empty them into a medium bowl.
In the bowl of the stand mixer fitted with the whisk beater, add the yolks to the sugar and beat on high speed for 5 minutes, or until very thick and fluffy and when the beater is raised the mixture falls in ribbons.
Detach the whisk from the mixer and use it to fold the pecan mixture and the coffee extract, if using, into the batter until evenly mixed. If you don’t have a second mixer bowl, scrape this mixture into a large bowl and thoroughly wash, rinse, and dry the mixer bowl and whisk beater to remove any trace of oil.
In the bowl of the stand mixer fitted with the whisk beater, beat the egg whites on medium speed until foamy. With the mixer off, add the cream of tartar, if using. Raise the speed to medium-high and beat until soft peaks form when the beater is raised. Gradually beat in the remaining 2 tablespoons of sugar and beat until stiff peaks form when the whisk is raised slowly. If not using cream of tartar, stop beating just before stiff peaks to prevent overbeating. The peaks should curve over slightly when the beater is raised.
Add about one-quarter of the meringue to the yolk mixture. With a large balloon whisk or the whisk beater, fold until completely incorporated. Gently fold in the remaining meringue in three parts. For the last addition, be sure there are no white streaks of meringue in the batter. Pour the batter into the prepared pan and, using a small off set spatula or silicone spatula, spread the surface evenly. The batter will fill the pan half full.
Bake for 30 to 40 minutes, or until the cake is springy to the touch when pressed very lightly in the center. An instant-read thermometer will register 185°F/85°C. In a 2-1/2-inch-high pan, the batter will have risen to the top of the pan.
Immediately invert the cake onto a wire rack that has been coated lightly with nonstick cooking spray. Leave it undisturbed until the pan feels completely cool to the touch. Reinvert the pan. Run a small metal spatula between the sides of the pan and the cake, pressing firmly against the pan, and remove the sides of the pan. Invert the cake onto a flat plate and remove the pan bottom and parchment. Reinvert it onto a serving plate. There will be a 3/8-inch depression to fill with coffee cream.
In a mixing bowl, combine the cream, sugar and espresso powder, if using. Refrigerate for at least 15 minutes (chill the mixer’s beaters alongside the bowl).
In a 1-cup heatproof glass measure, place the water and gelatin. Allow the mixture to soften for 5 minutes. Set the measure in a pan of simmering water and stir occasionally until the gelatin is dissolved. This can also be done in a microwave on high power, stirring once or twice.
Remove the cup from the water and cool the mixture to room temperature, about 7 minutes (it can be held longer, but should be covered to prevent evaporation). The gelatin must be liquid, but not warm when added to the cream.
Whip the cream mixture, starting on low speed, gradually raising the speed to medium-high as it thickens, just until traces of the beater marks begin to show distinctly. Add the gelatin mixture in a steady stream, whipping constantly. Add the vanilla and coffee extract, if not using the espresso powder, and whip just until stiff peaks form when the beater is raised. To avoid the risk of overwhipping, when almost stiff enough, remove the beaters and use them, or a whisk, to finish whipping by hand.
Immediately swirl the cream into the depression on top of the cake or scrape it into a bowl to pass on the side. The cake can be presented upside down or topside up. The completed cake can be refrigerated overnight and will keep at room temperature for several hours.
NOTE: If filling it with the coffee cream several hours ahead, it is especially important to add the gelatin to keep it from watering out.