In a medium saucepan, set over medium-low heat, add the butter, vegetable oil and milk. Heat the mixture between 100F to 110F and transfer it to a mixing bowl of a stand-up mixer. Mix in the egg and then sprinkle the yeast atop the mixture; allow it to stand until mostly foamy, about 10 minutes.
Meanwhile, in a large bowl, whisk together the flour, sugar and salt.
With the dough hook attached to the stand-up mixer, add a few spoonfuls of the dry ingredients to the wet. Turn the mixer to medium-low and continue adding the flour mixture, a spoonful at a time, until totally combined with the wet.
Turn the mixer to medium/high and knead the dough is smooth and sort of pulls away from the sides of the mixing bowl, 7 to 10 minutes. The dough will be slightly sticky. Grease a medium, clean bowl with cooking spray or olive oil. Form the dough into a ball (some will stick to your hands, don't worry) and place in the bowl and cover with plastic wrap. Transfer to the fridge to rise for 1 hour. (Putting in the fridge slows down the rising process.)
Line a baking sheet with parchment. Dust a work surface and the top of the dough in the bowl with a little bit of flour. Punch down the dough and transfer it directly to the floured surface. Roll it out, using a rolling pin, to a 1/2-inch thickness.
Use a 3-inch round cutter and 1 1/4-inch round cutter to cut out your doughnuts. Cut out as many as you can and reroll the scraps and cut out a few more.
Transfer the doughnuts and doughnut holes to the parchment lined baking sheet, spacing them about 2 to 3 inches apart. Spray the tops with a bit of cooking spray and lightly cover with plastic wrap. Allow to rise for 30 to 45 minutes, until almost quartered in size.