In a small bowl combine yeast, warm water and 1 teaspoon sugar. Mix and then set aside for about 10 minutes or until it starts to foam.
In the bowl of a stand mixer fitted with a paddle attachment, add salt, eggs and butter. Mix on low/med speed until light and fluffy. Add milk and 1 cup of flour. Mix until combined.
Add the yeast mixture to the bowl, mix until combined.
Switch to the dough hook attachment. Turn mixer to medium/low and add in enough of the remaining flour, 1/4 cup at a time, until the dough is barely sticky. Turn mixer to medium speed and knead dough until it is smooth and slightly elastic.
Turn the dough out into a large buttered/greased mixing bowl. Cover with plastic wrap; let rise until it doubled in volume.
Punch down dough and turn out onto a lightly floured surface. Roll the dough out to be about a ½-inch thickness, and cut into 3-inch circles (use a biscuit cutter, doughnut cutter, or a drinking glass). Place the dough circles on a parchment paper-lined baking sheet. Do not re-roll scraps, they can be cut into small strips or chunks for extra snacking, or discarded.
Cover; let rise until they have at least doubled in size and fluffy.
While the doughnuts rise, preheat cooking oil to 365°F degrees in either an electric fryer, or on a stove top in a heavy-bottomed Dutch oven.
Fry each doughnut until it is golden on the bottom, flip and fry until the other side is golden. Be very careful when transferring the doughnuts to the oil. If they are handled too much, they will collapse. (See note.) Place fried doughnuts onto paper towels to drain any extra grease.
Prepare coating: Add sugar to large bowl or small paper bag. Toss doughnuts in granulated sugar, and place on a cooling wrack to cool to room temperature.
Prepare the filling: Create a double boiler by bringing a pot of water to boil and placing a heat-proof mixing bowl on top. Whisk in lemon juice, zest, sugar and eggs. Continue whisking on the double boiler until it starts to thicken slightly. Drop butter into the mixture one tablespoon at a time.
Once the butter has been added, continue whisking until it is thick enough to coat the back of a spoon. Remove from heat, and place in the refrigerator for at least an hour, or until it has cooled completely.
Meanwhile, in a mixing bowl, whisk heavy cream and salt together until it forms stiff peaks. Carefully fold in the cooled lemon mixture. Place in the refrigerator for at least another 30 minutes.
Right before serving, cut small holes in the side of each doughnut with a paring knife, in the shape of an X. Place filling in a piping bag fitted with a round tip, a longer tip works best.
Fill each doughnut, and enjoy!