Fill a large wide pot with 3/4 inch of oil and heat to 360 degrees F.
Meanwhile, whisk together the flour, granulated sugar, baking powder and salt in a large bowl. Whisk together the butter, egg and 3/4 cup of the milk in a medium bowl. Add the wet ingredient to the dry and mix with a wooden spoon until the dough comes together. Lightly flour a work surface and turn out the dough; knead a few times until it comes together,
Divide the dough into 10 equal pieces. Roll each piece between your hands to make a ball and flatten each ball into a 3-inch-wide disk. Use your fingers to rip a hole in the center of each and gently spread each hole so that it is about 1 inch wide.
When the oil is hot, fry the donuts in batches, turning once, until deep golden brown all over, about 3 minutes per batch. Return the oil to 360 degrees F between batches. Transfer the donuts to a paper towel-lined tray as they are cooked; cool at least 15 minutes.
Meanwhile, warm the remaining 1/4 cup milk in a large microwave-safe bowl. Add the coffee and stir to dissolve. Add the confectioners' sugar and whisk until smooth. Add enough additional milk to make the glaze a nice coating consistency.
Dip one side of each donut in the glaze and place on a plate. Sprinkle with the rainbow sprinkles. Serve warm or at room temperature.