Mix the all-purpose flour, granulated sugar, instant yeast, and fine salt. Add the warm whole milk, Greek yogurt, melted butter, and vanilla extract, then knead for 7 to 9 minutes until the dough is smooth, soft, and slightly tacky.
Place the dough in a lightly greased bowl, cover, and let it rise until doubled, about 75 to 90 minutes.
Roll the dough to 1/2-inch thickness and stamp out 3-inch rounds. Arrange the rounds on parchment, cover loosely, and let them rise for 35 to 45 minutes until puffy.
For the pastry cream, whisk 1/2 cup of the whole milk with the cornstarch. Add the remaining milk, granulated sugar, and fine salt, then cook over medium heat while whisking until thick and bubbling. Cook for 1 minute more, remove from the heat, and whisk in the butter and vanilla extract. Chill until cold.
Heat the vegetable oil in a deep heavy pot to 350°F. Fry the donuts in small batches for 60 to 90 seconds per side until deep golden and cooked through.
Transfer the fried donuts to a rack and let them cool completely before filling.
For the glaze, heat the heavy cream until steaming and pour it over the chopped semisweet chocolate. Add the light corn syrup, let stand for 2 minutes, then stir until smooth and glossy.
Whisk the chilled pastry cream until smooth and transfer it to a piping bag fitted with a small round tip. Make a small opening in the side of each donut and pipe in enough pastry cream to make the donut feel noticeably heavier.
Dip the top of each filled donut into the chocolate glaze and place it on a rack. Let the glaze set for 15 to 20 minutes before serving.
