In each glass combine 1/4 cup each of the milk and strawberry syrup. Put 2 or 3 scoops of the ice cream into each glass and fill the glasses with the club soda or seltzer. Divide the remaining syrup among the sodas and garnish each soda with a strawberry.
In the food processor or in a blender in batches puree the strawberries, transfer the puree to a stainless steel or enameled saucepan, and stir in the sugar and water. Bring the mixture to a boil over moderate heat, stirring, and simmer it for 1 minute. Force the mixture through a fine sieve set over a bowl and chill the syrup, covered, for at least 3 hours.