In a large bowl, mix sourdough discard, melted butter, warm milk, sugar, salt, baking powder, and flour. Stir until a soft dough forms.
If sticky, add more flour. Knead on a floured surface for 2-3 minutes until smooth. Cover and let rest for 15-20 minutes.
Roll dough into a 12x9-inch rectangle. Spread softened butter on the dough, leaving a border. Mix brown sugar and cinnamon, then sprinkle evenly over the dough.
Cut dough into 1-inch strips, then slice into 1-inch squares (about 80-100 pieces). Grease a muffin tin and place 2-3 dough squares into each cup, pressing down slightly.
Preheat oven to 375°F (190°C). Bake for 10-12 minutes, or until golden and puffed. Cool in the pan for 5 minutes, then transfer to a wire rack.
Beat cream cheese, powdered sugar, and vanilla extract until smooth. Add milk, one tablespoon at a time, until the glaze reaches a drizzling consistency.
Drizzle the cream cheese glaze over the warm bites and serve immediately, or store in an airtight container for up to 2 days.