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Grease an 8-inch square baking pan.
Place sugar and cocoa powder in a medium saucepan; pour in milk and stir until blended. Set heat to medium-high and stir constantly until mixture comes to a boil.
Reduce heat to low and place a candy thermometer in the pan. Let the mixture simmer without stirring until the temperature reaches 238 degrees F (114 degrees C) when measured with a candy thermometer, about 10 minutes.
If you don't have a candy thermometer, drop a small amount of the mixture into cold water; if it forms a soft ball that flattens when removed from the water and placed on a flat surface, it's ready.
Remove from the heat. Allow to cool to 110 degrees F (43 degrees C), 50 to 70 minutes. Do not disturb fudge as it's cooling.
Add butter and vanilla to the fudge. Beat with a wooden spoon until well incorporated and fudge loses its sheen; do not under beat.
Pour fudge into prepared pan and press to flatten. Let cool at room temperature or in the refrigerator before cutting into 1-inch squares.