Prepare the Dough:

In a small bowl, combine the warm milk, 1 tablespoon of granulated sugar, and active dry yeast. Stir gently and let it sit for about 5 minutes, or until it becomes frothy and bubbly—this indicates that the yeast is activated.
In a large mixing bowl, combine the all-purpose flour, remaining granulated sugar, and salt. Add the melted butter, egg, and the activated yeast mixture. Mix until a sticky dough forms.
Transfer the dough to a floured surface and knead for 5-7 minutes until smooth and elastic. If the dough is too sticky, add a little more flour, 1 tablespoon at a time.
Place the dough in a lightly greased bowl, cover with a clean kitchen towel, and let it rise in a warm place for 1-2 hours, or until it has doubled in size.
Form and Fry the Donuts:

Once the dough has risen, punch it down and transfer it to a floured surface. Roll out the dough to about 1/2 inch (1.25 cm) thickness.
Use a donut cutter or a round cutter to cut out donut shapes. Gather the scraps, re-roll, and cut out additional donuts.
Place the cut-out donuts on a lightly floured surface, cover with a towel, and let them rise for another 30-45 minutes, or until they are puffy and doubled in size.
Heat oil in a deep fryer or a large, heavy-bottomed pot to 375°F (190°C). Fry the donuts a few at a time, about 1-2 minutes per side, until they are golden brown.
Remove the donuts with a slotted spoon and drain on paper towels.
Prepare the Strawberry Cheesecake Filling:

In a medium bowl, beat the softened cream cheese, powdered sugar, and vanilla extract together until smooth and creamy.
Gently fold in the strawberry jam until well combined. Transfer the mixture to a piping bag fitted with a round tip.
Fill the Donuts:

Once the donuts have cooled completely, use a small knife to create a hole in the side of each donut.
Insert the piping bag tip into the hole and pipe the strawberry cheesecake filling into each donut until filled.
Make the Glaze:

In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth.
Drizzle the glaze over the filled donuts, or dip the tops of the donuts into the glaze for a more uniform coating.
Serve and Enjoy:

Serve these Strawberry Cheesecake Stuffed Donuts immediately for the best texture and flavor. Store any leftovers in the refrigerator and enjoy within 2-3 days.