Place the strawberries in a large mixing bowl. Sprinkle with the sugar and a good squeeze of lemon juice and toss. If using, drizzle over the Cointreau and toss again. (To save time, you can do this before preparing the other components above.)
Let rest for 10 minutes, tossing from time to time, until the sugar has completely melted into the strawberries to form a kind of strawberry syrup.
Transfer the strawberries to a blender and purée until smooth.