Preheat fryer with peanut oil to 350 degrees F.
Separate pre-cut biscuits. Make a hole in the middle with your hand and sculpt dough into a doughnut shape. Place doughnuts into fryer and cook for 4 to 5 minutes, flipping to fry both sides. Place doughnuts aside onto a paper towel lined plate to remove excess oil.
Melt the chocolate and butter in a saucepan over very low heat. Stir in the sugar, alternating with the evaporated milk, blending well. Raise the heat to medium and bring to a boil, stirring occasionally. Lower the heat and cook, stirring, until the mixture becomes thick and creamy, about 8 minutes. Stir in the vanilla then remove the pan from the heat.
Stick a fork through the doughnut holes and dip the doughnuts immediately into the chocolate mixture before the chocolate starts to cool. Place the doughnuts on a plate and serve when the chocolate on the doughnuts sets, about 5 minutes.