Grease an 8-inch square baking pan and line with 2 pieces of parchment paper about 14 inches long; allow the ends of the paper to hang over the sides of the pan by several inches.
Stir evaporated milk, light brown sugar, white sugar, unsalted butter, and salt in a heavy saucepan over medium heat until the mixture forms a thick, smooth syrup.
Bring to a boil, reduce heat to medium-low, and cook until a candy thermometer inserted into the center of the boiling syrup without touching the bottom of the pan reads 238 degrees F (112 degrees C), 20 to 30 minutes. Stir often. Candy temperature will rise slowly at first, then more quickly as temperature increases; watch carefully.
Remove saucepan from heat and stir in the Irish cream liqueur and vanilla extract until thoroughly combined.
Scrape the mixture into the work bowl of a large stand mixer fitted with beaters; set the machine on low speed and mix in confectioners' sugar in 3 additions, allowing the sugar to blend in completely before adding the next addition. Scrape down the bowl frequently and mix fudge until completely smooth, about 2 minutes.
Scrape fudge into the prepared baking pan and place on a level surface to cool and set up, 3 to 5 hours.
Use the parchment paper ends as handles to lift the fudge out of the baking pan before slicing into cubes for serving.
