Pour warm water into the bowl of a stand mixer with the dough hook attachment. Sprinkle the yeast over the water and allow to proof for 5 minutes. Meanwhile, heat up the milk and butter in a small pot, until the butter is melted. Once melted, remove from the heat and allow to cool to below 110 degrees Fahrenheit. If the liquid is too hot, it can kill the yeast.

Add the eggs, sugar, salt, warm milk mixture, and half of the flour to the bowl. Turn on the mixer on a slow speed and mix until well combined. Turn up the speed to medium and add in the remaining flour. Mix until it comes into a ball of dough. Dump out the dough onto a floured surface and knead by hand for about 4 minutes, until smooth.

Place dough in a large greased bowl, cover with a linen towel and allow to rise for 1 hour or until doubled in size.

Roll out the dough on a floured surface to 1 inch thick. Cut the doughnuts into circles using a doughnut cutter or 4 inch cookie cutter. Set on a floured baking sheet, cover lightly with a tea towel, and let rise for 30 minutes.

Preheat the oil in a large pot to 350 degrees Fahrenheit. Gently place the doughnuts into the oil, 1 to 2 at a time. Cook for 1 minute per side then transfer to a cooling rack.

In a large bowl, combine cinnamon and sugar and stir to combine. Set aside.

Combine sugar, flour, cornstarch, cardamom and a pinch of salt in a small saucepan. Over medium heat, whisk in eggs, cream and vanilla paste. Continue cooking, whisking almost constantly, until the mixture just begins to boil and thickens, about 5 minutes. The mixture should be stiff, which you can test by coating the back of a spoon and drawing your finger through this coating, the line should hold its shape. Remove from the heat and stir in softened unsalted butter. Then, strain through a fine-mesh sieve and cool completely before using.

While the pastry cream is cooling, toss the warm doughnuts in the cinnamon sugar mixed and set on a racked sheet tray.

Add the cooled cream to a pastry bag fitted with a star tip and set aside. Then, using a chopstick or a wooden skewer, gently make a hole in the side of the doughnut, being careful not to puncture all of the way through to the back. Using your finger, wiggle it into the hole of the doughnut to make more space for the pastry cream. Then insert the tip of the pastry bag and squeeze the cream into the doughnut until it just begins to poke out of the front of the hole. Transfer to a serving platter and enjoy!