Line a 9-by-5-inch loaf pan with foil, leaving a 2-inch overhang on all sides.
Lightly brush the foil with butter. Lightly brush the bowl of a stand mixer or another large metal bowl with butter. Add 2 tablespoons cut-up butter, the chocolate, extract and salt; set the bowl aside.
Heat the half-and-half in a medium pot over medium-high heat until hot but not boiling. Stir in the sugar and corn syrup and bring to a boil, stirring constantly and scraping the bottom of the pot with a wooden spoon; boil, stirring, until the sugar is dissolved, about 3 minutes. Reduce the heat to low; run a pastry brush dipped in warm water along the sides of the pan to dissolve any sugar crystals clinging to the pan. Clip a candy thermometer to the pan. Simmer the mixture, undisturbed, until the thermometer registers 234 degrees F to 238 degrees F, 20 to 30 minutes, watching the temperature closely as cooking times may vary. (This is the "soft-ball stage." To test for doneness, drop some of the mixture into cold water; you should be able to roll it into a soft ball.)
Quickly pour the sugar mixture over the chocolate-butter mixture in the bowl (do not scrape the sides or bottom of the saucepan in case any sugar crystals formed). Clean the candy thermometer, then clip it to the bowl. Let the chocolate mixture cool, undisturbed, until it registers 110 degrees F to 115 degrees F, about 1 hour. (For faster cooling, set the bowl in a larger bowl of cold water.)
Brush the paddle attachment or mixer beaters with butter. Beat the mixture on medium speed until just incorporated, about 1 minute. Reduce the mixer speed to low and beat until the fudge just begins to lose its sheen and hold its shape, 5 to 15 minutes. Do not overmix or the fudge will become hard. Use a buttered rubber spatula to scrape the fudge into the prepared pan and pat into an even layer; smooth the top.
Press one or more toppings into the fudge, if desired (about 3/4 cup total). Let sit at room temperature for 1 hour, then score the fudge into small squares with a knife. For the best texture, let sit at least 1 more hour before slicing. Wrap leftover fudge in plastic wrap and store in an airtight container for up to 1 week at room temperature, or up to 2 weeks in the refrigerator. Bring to room temperature before serving.
Toppings: Crushed sandwich cookies, crushed graham crackers, white chocolate chips, sprinkles, peanut butter drizzle, or mini marshmallows.