Step 1: In a large bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
Step 2: In a separate medium bowl, whisk together the warm milk, granulated sugar, egg, melted butter, and vanilla extract until well combined.
Step 3: Pour the wet ingredients into the dry ingredients. Mix with a spoon or spatula until just combined. Do not overmix. Gently fold in ¼ cup of finely diced fresh strawberries. The batter will be thick.
Step 4: For the cheesecake filling, beat the softened cream cheese, powdered sugar, and vanilla extract together in a small bowl until smooth and creamy. Transfer to a piping bag with a small round tip, or a ziptop bag with the corner snipped.
Step 5: Heat about 2-3 inches of vegetable oil in a large heavy-bottomed pot or Dutch oven to 350-375°F (175-190°C).
Step 6: Spoon a tablespoon of donut batter into the hot oil. Quickly pipe a small dollop of cheesecake filling into the center of the batter, then top with another tablespoon of donut batter to enclose the filling. Repeat for 2-3 donuts at a time, being careful not to overcrowd the pot.
Step 7: Fry the donuts for 2-3 minutes per side, or until golden brown and cooked through. Remove with a slotted spoon and place on a wire rack lined with paper towels to drain excess oil. Allow to cool slightly.
Step 8: For the strawberry glaze, blend the fresh strawberries in a food processor or with an immersion blender until smooth. In a bowl, whisk together the strawberry puree, powdered sugar, and 1 tablespoon of milk/lemon juice until smooth. Add more milk/lemon juice a teaspoon at a time if the glaze is too thick.
Step 9: Once the donuts have cooled slightly, dip each donut into the strawberry glaze, allowing excess to drip off. Place back on the wire rack to set. Serve warm or at room temperature.