Combine milk, yeast, sugar, egg, flour, butter, and salt in a stand mixer fitted with a dough hook. Knead on medium-high speed until smooth and elastic, about 8 minutes. Cover and let rise at room temperature until doubled, about 1 hour.
Mix butter, brown sugar, and cinnamon into a smooth paste and transfer to a piping bag.
Divide dough into 2-ounce/60-gram portions. Roll each into a rectangle, pipe filling along one long edge, roll tightly, and gently stretch into an even rope. Shape into a loop, twisting the ends together to seal.
Place donuts on individual squares of parchment, cover, and let rise until doubled, about 1 hour.
Heat 1½ quarts/liters canola oil to 330°F/165°C. Remove parchment and fry donuts until deep golden, 1½ to 2 minutes per side. Transfer to paper towels to cool slightly.
Whisk melted butter and powdered sugar until smooth, then mix in cream cheese. Drizzle glaze over warm donuts and serve.