Summarize:

Place 2 cups (400 grams) of fresh, sliced strawberries, ½ cup (115 grams) of granulated white sugar, and ½ cup (120 milliliters) of water into a saucepan. Bring the mixture to a boil over medium-low heat, then reduce the heat to low, and simmer for 10 minutes. Pour the strawberry mixture into a blender or food processor, and purée until smooth, about 10 seconds. From time to time, pause the blender/food processor, and use a rubber spatula to push any lumps or clumps down towards the bottom. The smoother the purée is, the smoother your buttercream will be. Pour the syrup into a bowl, and cover it with a sheet of plastic wrap. Put it into the fridge, and leave it there for at least 20 minutes. Keep the leftover strawberry syrup in a jar or bottle with a tight-fitting lid, and use it within one week. You can use it on top of ice cream, waffles, pancakes, and other desserts. You can do this using a handheld mixer, a stand mixer, or a food processor fitted with whisks. Be sure to scrape the bottom and sides of the bowl. The more sugar you add, the stiffer your buttercream will be. Keep adding the sugar and strawberry syrup a little bit at a time until you have used up all of your sugar. Remember, you are only using 4 tablespoons (60 milliliters) of the strawberry syrup. Beat it in using a high speed setting for 20 seconds. This will lighten the frosting. Vanilla extract can change the color of frosting slightly. If this bothers you, use the clear, colorless kind. Keep adding the half-and-half until the buttercream holds its shape. You may not need to use all of it. Use a high speed setting, and beat it for only 20 seconds. This will help cut the sweetness a little while lightening the frosting up further. If you need to use refrigerated buttercream, let it sit on the counter until it reaches room temperature, then beat it on low speed until it turns smooth. If you'd like a thicker, stiffer buttercream, refrigerate the buttercream for 40 to 45 minutes. Be sure to cover it with a sheet of plastic wrap so that it doesn't dry out.
Cook the strawberries, sugar, and water in a medium-sized saucepan for 10 minutes. Purée the strawberry mixture. Chill the strawberry syrup in the fridge for at least 20 minutes. Set aside 4 tablespoons of the strawberry syrup, and store the rest in the fridge. Beat the butter and ½ cup (65 grams) of the powdered sugar. Add 1 tablespoon (15 milliliters) of the strawberry syrup and beat again. Add the vanilla extract. Beat in the half-and-half, 1 tablespoon (15 milliliters) at a time. Beat in the pinch of salt. Use the frosting right away, or cover it and keep it in the fridge for up to three days.