To candy the bacon, preheat the oven to 400F (200C).
Lay the strips of bacon on a baking sheet lined with a silicone mat or aluminum foil, shiny side down.
Sprinkle 1½-2 teaspoons of brown sugar evenly over each strip of bacon, depending on length.
Bake for 12-16 minutes. Midway during baking, flip the bacon strips over and drag them through the dark, syrupy liquid that’s collected on the baking sheet. Continue to bake until as dark as mahogany. Remove from oven and cool the strips on a wire rack.
Once crisp and cool, chop into little pieces. (Bacon Bits can be stored in an airtight container and chilled for a day or so, or stored in the freezer a few weeks ahead.)
To make the ice cream custard, melt the butter in a heavy, medium-size saucepan. Stir in the brown sugar and half of the half-and-half. Pour the remaining half-and-half into a bowl set in an ice bath and set a mesh strainer over the top.
In a separate bowl, stir together the egg yolks, then gradually add some of the warm brown sugar mixture to them, whisking the yolks constantly as you pour. Pour the mixture back into the saucepan.
Cook over low to moderate heat, constantly stirring and scraping the bottom with a heatproof spatula, until the custard thickens enough to coat the spatula.
Strain the custard into the half-and-half, stirring over the ice bath, until cool. Add rum and vanilla.
Refrigerate the mixture. Once thoroughly chilled, freeze in your ice cream maker according to the manufacturer’s instructions. Add the bacon bits during the last moment of churning or stir them in when you remove the ice cream from the machine.