Place the strawberries, sugar, vanilla and lemon juice in a medium saucepan over medium-high heat. Bring to a gentle boil, stirring constantly, and cook until the strawberries are very soft, a good 5 minutes. Turn off the heat and use a potato masher to completely smush them to smithereens. Pour the mixture into a fine-mesh strainer placed over a bowl. Use a spoon to stir the fruit so the liquid is forced through. Continue until almost all of the liquid is in the bowl. Set the pulp aside.
Skim as much foam off the sauce as you can. Pour the skimmed sauce back into the same saucepan, add 1/4 cup of the pulp back in and boil again for 3 minutes over medium-high heat. Skim off any more foam that has formed on the surface, then allow the sauce to cool slightly before pouring it into a pitcher or jar. Refrigerate it to thicken and serve it cold.