Make the filling: In a small bowl, mix softened cream cheese, powdered sugar, and vanilla extract until smooth. Scoop into ½ teaspoon portions and freeze on a parchment-lined tray for about 30 minutes until firm.
Prepare the donut hole batter: In a mixing bowl, whisk together flour, baking powder, salt, and sugar. In another bowl, mix milk, egg, melted butter, and vanilla. Combine wet and dry ingredients until a thick batter forms.
Shape the donut holes: Scoop about 1 tablespoon of dough, flatten slightly in your hand, and place a frozen cream cheese portion in the center. Wrap the dough around it and roll into a smooth ball. Repeat with remaining dough.
Bake or fry:
To Bake: Preheat oven to 375°F (190°C). Place donut holes on a lined baking sheet and bake for 12–14 minutes until golden.
To Fry: Heat oil in a deep pan to 350°F (175°C). Fry in batches for 2–3 minutes, turning, until golden brown. Drain on paper towels.
Coat and fill: While warm, brush or dip each donut hole in melted butter, then roll in granulated sugar. Use a piping bag to inject a small amount of strawberry jam into the center (or simply drizzle over the top).
Serve warm or at room temperature.