Combine warm milk, yeast, and 1 tablespoon sugar in a small bowl; let sit 5-10 minutes until foamy.
In a large bowl, whisk flour, salt, and remaining sugar; add yeast mixture, melted butter, and egg; mix to form shaggy dough.
Knead dough on floured surface for 8-10 minutes until smooth; place in greased bowl, cover, and let rise 1-1.5 hours until doubled.
Punch down dough, roll to 1/2 inch thickness, and cut into donuts; let rest 20-30 minutes.
Heat oil to 350-360°F; fry donuts 1.5-2 minutes per side until golden brown; drain on paper towels.
Beat cream cheese, powdered sugar, and vanilla until smooth; fold in strawberry jam gently to create swirls.
Whisk powdered sugar, milk, and vanilla until glaze is smooth and pourable.
Inject cheesecake filling into warm donuts; dip or drizzle tops with glaze; let set 15-20 minutes before serving.