Once the dough has rested and is firm and cold, roll it down to 1.25cm/.5in using either a sheeter or a rolling pin. Let the dough relax for 30 minutes in the refrigerator.
Using a 7.5-cm/3-in doughnut cutter dipped in flour, cut out as many doughnuts as possible and place them on a sheet pan lined with silicone paper. Proof the dough at room temperature in a large plastic bag for 2 hours or until doubled in size. You will only need 4 doughnuts and 4 doughnut centers for 4 cakes. Eat the remaining doughnuts (or see the Note).
Set a fryer filled with peanut oil to 180°C/355°F.
Fry 4 doughnuts at a time until golden brown, about 30 seconds on each side.
Once fried, toss them in granulated sugar as soon as they come out of the fryer to completely coat them; repeat this procedure with the doughnut centers. Keep in mind that these will fry faster. Turn them constantly in the frying oil so that they color evenly and stay round. This takes about a total of 30 seconds.
Let them cool on a sheet pan at room temperature. Freeze the doughnuts to assemble the cakes. Save the doughnut centers to garnish the cake.