Combine milk, cream and Baileys in a medium-sized saucepan over medium-low heat. Stirring occasionally, heat until the mixture reaches a temperature of 170°F and bubbles start to form around the edges.
While the cream mixture is heating, whisk the egg yolks together until smooth in a heatproof bowl. Gradually whisk in the sugar until well incorporated and the mixture is thick and a pale yellow color. Temper the egg yolks by very slowly pouring the hot cream mixture while whisking constantly. Pour this mixture back into the saucepan and heat, stirring frequently, on low heat to a temperature of 185°F. The mixture should be slightly thickened, enough to coat the back of a spoon. It should not come to a boil.
Pour the custard mixture through a fine mesh strainer into a clean bowl in an ice bath to cool the mixture completely, stirring constantly. Cover and refrigerate at least 4 hours or overnight.
Remove the custard from the refrigerator and stir in the vanilla extract and salt. Pour the mixture into an ice cream machine and churn for about 20 minutes (follow your ice cream maker directions).
Transfer ice cream to airtight container and freeze until solid, about 4 hours.
Notes
Use the Best Ingredients - when making ice cream you want to use the best ingredients possible so use free-range eggs and organic milk and cream. Don't use imitation vanilla extract either. Use the real thing or make your own homemade vanilla extract.
Don't Cut the Fat and Calories - ice cream is not the dessert that you should be cutting the fat. It depends on the high-fat content to create a creamy texture. If you go lower fat, it will cause an icy ice cream. If you're on a diet, just have a smaller scoop!
Add Extract at Right Time - if you add the vanilla extract while the custard is too hot, you won't get the actual flavor as it will cook out slightly. Just wait until right before you pour the custard into the ice cream maker to add it.
Chill the Custard - you don't necessarily need to chill the custard overnight, but you want to chill until it is as cold as possible.
Use Frozen Bowl - make sure your ice cream bowl has been frozen for at least 24 hours, otherwise you aren't giving ice cream a great chance of freeze.
Start Ice Cream Motor First - do not pour the custard in the frozen bowl before starting the ice cream maker. It will start freezing upon contact. You want the motor running and slowly pour the custard in.
Don't Overchurn - always error on the side of caution when it comes to churning your ice cream. You do not want it to be hard. Remember it will continue to freeze once you place in the freezer.