1. Combine the flour, yeast, sugar, and salt in the bowl of a stand mixer and briefly mix using the dough attachment.
2. Gradually add warmed milk, eggs, and softened butter to the dry ingredients and mix on medium speed for 8-15 minutes until the dough is soft, elastic, and passes the windowpane test.
3. Transfer the kneaded dough to a lightly greased bowl, cover with a damp cloth, and let it proof for 45-60 minutes until it doubles in size.
4. Heat whole milk (and optional heavy cream) with vanilla extract in a saucepan over medium heat until it begins to steam, avoiding boiling.
5. In a separate bowl, whisk egg yolks, sugar, flour, and cornstarch until smooth, then gradually introduce the warm milk mixture while continuously whisking.
6. Return the mixture to the saucepan and whisk over medium heat until it thickens and boils. Remove from heat, stir in butter, strain, and let the pastry cream cool.
7. Roll out the proofed dough to about ½ inch thickness on a lightly floured surface. Cut into donut shapes using circular cutters and allow them to rise for 15-20 minutes.
8. Heat oil in a deep fryer or heavy-bottomed pot to 350°F (175°C) and fry the donuts in batches, cooking each side for about 2 minutes until golden brown. Remove and drain on paper towels.
9. Mix sugar, water, and a pinch of salt in a light-colored saucepan, cooking over medium heat until the sugar turns a deep amber color, avoiding stirring during caramelization.
10. Carefully dip the tops of the warm donuts into the hot caramel to create a glossy, crispy shell, then allow the caramel to cool and set.
11. Puncture a hole in each caramel-coated donut and use a piping bag to fill them with the chilled vanilla pastry cream, ensuring even distribution.