Make the Dough
In a large bowl, combine the flour, yeast, sugar, and salt. Mix until well combined.
Add the warm milk, eggs, and softened butter. Mix until a sticky dough forms.
Knead the dough on a floured surface for 8-10 minutes or until smooth and elastic. Alternatively, use a stand mixer with a dough hook for 5-6 minutes.
Place the dough in a greased bowl, cover with a clean towel, and let it rise in a warm place for 1-1.5 hours or until doubled in size.
Shape the Donuts
Roll out the dough on a lightly floured surface to about ½ inch thick. Use a donut cutter or round cutters to cut out the donuts.
Place the cut donuts on a baking sheet lined with parchment paper. Cover and let them rise again for 30 minutes until puffy.
Fry the Donuts
Heat oil in a deep fryer or pot to 350°F (175°C). Carefully fry the donuts in small batches, cooking for about 1-2 minutes per side until golden brown.
Remove and drain on a wire rack lined with paper towels. Let the donuts cool completely.
Prepare the Filling
In a mixing bowl, beat the cream cheese and powdered sugar until smooth. Add the diced strawberries (or preserves) and vanilla extract, mixing until well combined.
Transfer the filling to a piping bag fitted with a large round or star tip.
Fill the Donuts
Use a skewer or knife to create a hole in the side of each donut. Insert the piping tip and fill the donuts with the strawberry cheesecake filling until plump.
Make the Glaze
In a bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth. Adjust the milk for a thicker or thinner glaze.
Glaze the Donuts
Dip the tops of the filled donuts into the glaze and place them back on the wire rack to set.