Heat milk, sugar, and salt in a medium saucepan over medium heat. Stir constantly, until the mixture just reaches the point of simmer. Do not let boil.
In a separate medium bowl, lightly whisk egg yolks. Pour about half of the warm milk into the bowl with the egg yolks, whisking while you do so. Pour the egg/milk mixture back into the saucepan with the rest of the milk.
Cook mixture over low heat, stirring constantly, until it thickens enough to coat the back of the spoon and/or reaches 160 degrees.
Pour the heavy cream into a separate large bowl. Strain the egg/milk mixture into the heavy cream and stir to combine. Stir in Baileys Irish Cream. Cover with plastic wrap and place in the fridge for at least 4 hours.
Once mixture has been sufficiently chilled, process in ice cream maker according to manufacturer’s directions.
Scoop ice cream in airtight containers and place in the freezer for another 4+ hours, but ideally overnight.