In a saucepan over medium-high heat, stir together strawberries, sugar, and 1 1/4 cups water.
Let the mixture come to a boil.
Boil, stirring every couple minutes, for 10 minutes. (Be sure to reduce the heat if the syrup looks like it will boil over.)
Stir together 2 Tbsp water and 1 Tbsp cornstarch and pour it into the boiling syrup. Continue cooking, stirring frequently, until the syrup thickens. (This should take 2 to 4 minutes more. If you’re going to leave the strawberry pieces in the syrup, you can choose to continue cooking the syrup for a few minutes longer to let the strawberry pieces reach a softer texture.)
Optional: If you want to strain out the strawberries, pour the syrup through a sieve. Do not press on the strawberries (this will cause your syrup to be cloudy).
Let the syrup cool completely.
Refrigerate for up to 3 weeks.