Prep dish. Butter a square dish (7x7, 8x8, or 9x9) and set aside.
Combine all ingredients except butter and extracts. In a 2-quart saucepan, combine the sugar, half-and-half, salt, and corn syrup. Whisk to combine and bring to a boil over medium-low heat.
Heat to 240°F. Cook the mixture without stirring until it reaches 240°F, as measured by a candy thermometer. It is common for the mixture to heat up quickly at first, then slow down in the 230s. Be patient. Once the temperature is reached, immediately remove the pan from the heat.
Add butter and extracts. Let the sugar mixture sit for a minute to stop bubbling. Place the butter, peppermint extract, and vanilla extract on top, but do NOT stir. Allow the mixture to cool at room temperature for about 20-30 minutes until the side of the pan is warm, but not hot, to the touch.
Beat until firm. Using a hand mixer on medium-low, beat the fudge mixture for 1-3 minutes until it just starts to firm up and lose its shine. Be careful not to overbeat.
Pour into a prepared dish. Quickly pour or scoop the fudge into a buttered 8x8-inch or 9x9-inch dish. Sprinkle crushed peppermint candies or candy canes on top.
Cool and cut. Let the fudge cool completely for about 20 minutes, then cut into 1-inch squares and enjoy!