Preheat the oven to 350 degrees F. Dissolve the instant coffee in a bowl with 1/3 cup warm water. Cook the sugar in a medium skillet over medium-high heat, stirring occasionally, until it begins to melt. Continue cooking, swirling the pan but not stirring, until the sugar turns amber, about 7 minutes. Remove from the heat (don't worry if a few lumps of sugar remain). Carefully stir in the coffee mixture until smooth. Pour the caramel into an 8-inch-square baking pan; tilt the pan to coat the bottom and halfway up the sides. Let cool until the caramel is no longer sticky, about 10 minutes.
Lay the cake halves, cut-side down, next to each other on top of the caramel in the prepared pan. Poke the cake all over with a fork.
Whisk the eggs and yolks in a bowl until foamy. Slowly whisk in the condensed milk and evaporated milk until smooth. Pour the egg mixture over the cake, gently pressing on the cake so it soaks up the liquid. Cover the pan with foil and bake until set, 50 minutes to 1 hour. Transfer to a rack, remove the foil and let cool 30 minutes.
Run a knife around the edge of the pan to loosen the cake, then invert it onto a plate and unmold, letting the caramel drip onto the plate. Refrigerate until cold, about 2 hours. (The cake can be refrigerated, covered, up to 3 days.)
Beat the cream with a mixer until soft peaks form. Slice the cake; top with the whipped cream.