Add egg, milk, instant yeast, all purpose flour, granulated sugar, and salt to a large mixing bowl. Stir with a spatula until roughly combined. (The dough is very sticky now, so do not knead the dough with your hands yet.)
Cover the bowl with food wrap, rest the dough for 15 minutes. (The dough will be less sticky after the resting.)
Remove the food wrap, add the softened butter to the dough. Use your hands to knead the dough for about 3 minutes until the butter is fully absorbed. Then use the "slap & fold technique" (see note 2) to knead the dough for 5 more minutes until smooth.
Place the dough in the mixing bowl, cover the bowl with food wrap, rise the dough in a warm place for 1½ hour, or until it's doubled in size. (Ideally use the proof/warm mode of your oven. It might take the dough 2 hours to rise under room temperature during winter time.)
Transfer the dough to a nonstick pastry mat, punch out the air from the dough by making a fist with your hands and punching the dough to deflate it. Evenly divide the dough into 10 pieces (about 55g each). Knead each piece of the dough for 1 minute to remove air pockets, then roll into 10 balls.
Cut a sheet of parchment paper into 10 pieces of 5''x5'' squares, arrange them on a baking pan. Place one dough ball on each piece of the parchment paper. Cover the baking pan with food wrap. Proof the dough balls in a warm place for 1 hour, or until they are doubled in size. (Ideally use the proof/warm mode of your oven. It might take the dough 2 hours to rise under room temperature during winter time.)
Gently flatten each dough ball into 3'' diameter x 1'' thick oblate spheroid.
Heat a pot of oil (about 2 inch depth of oil) to 330°F. Carefully place the dough with the parchment paper into the hot oil, hold one corner of the paper to lift it away. Deep fry one side of the donut for 2 minutes, flip and fry the other side for 2 minutes until golden. (Use medium low heat to keep the oil temperature stay between 320°F - 340 °F.)
Remove the donuts with a pair of tongs, place the fried donuts on a cooling rack (put a large baking pan underneath) to drain oil.