Combine and heat. In a 2-quart saucepan, mix the sugar, salt, half and half, and corn syrup. Then, stir with a whisk until the mixture is smooth. Place the saucepan over medium-low heat and bring the mixture to a gentle boil.
Cook the mixture until 240°F. Attach the candy thermometer to the side of the saucepan to monitor the temperature. Let the mixture cook undisturbed (avoid stirring) until it reaches 240°F. Once it reaches this temperature, promptly remove the saucepan from the heat.
Add the butter and vanilla – do not stir. Allow the mixture to sit until it stops bubbling. Then, add the butter and vanilla on top, but do not stir or disturb the mixture. Let it cool at room temperature for about 20 minutes. The mixture is ready when the side of the pan is warm but no longer hot to the touch.
Beat the fudge until it thickens. Using a hand mixer at medium to low speed, beat the fudge for 1-3 minutes or until it begins to thicken and lose its shine. Be careful not to overbeat, as this can make the fudge too firm to pour.
Pour, cool, and cut. Quickly pour the fudge into a buttered dish, smoothing it out if needed. Let it set for about 20 minutes at room temperature. Once set, cut the fudge into one-inch squares and enjoy!