In a bowl, combine 1/2 cup of warm milk, 2 teaspoons of active dry yeast, and 1 tablespoon of granulated sugar. Let sit for 5-10 minutes until foamy.

In a large bowl, whisk together 3 cups of all-purpose flour, 1/2 teaspoon of salt, and 1/4 cup of granulated sugar. Pour in the yeast mixture, melted butter, and egg. Mix until a shaggy dough forms and knead until well combined.

Transfer the dough to a floured surface and knead for 8-10 minutes until smooth. Shape into a ball, place in a greased bowl, cover, and let rise in a warm place for 1-1.5 hours until doubled in size.

Once risen, roll the dough to 1/2 inch thickness and cut out donuts. Let rest for 20-30 minutes while heating oil to 350-360°F. Fry each donut for 1.5-2 minutes per side until golden brown.

In a medium bowl, beat together 8 ounces of softened cream cheese, 1/2 cup of powdered sugar, and 1 teaspoon of vanilla extract until smooth. Fold in 1/4 cup of strawberry jam.

Whisk together 1 cup of powdered sugar, 2 tablespoons of milk, and 1/2 teaspoon of vanilla extract until smooth. Adjust consistency with more milk or sugar as needed.

Inject the cheesecake filling into each cooled donut using a piping bag. Dip the tops into the glaze and let set for 15-20 minutes before serving.