Combine remaining strawberries, sugar, lemon zest and 1/4 cup water in a medium saucepan. Bring to a boil. Cook, stirring often, until berries are softened, about 5 minutes.
Transfer to a blender or food processor and pulse once or twice to make a chunky sauce.
Transfer to a bowl and refrigerate until chilled.
Combine ice cream and reserved strawberries in a large bowl and mash ice cream with back of a large spoon until strawberries and ice cream are well combined. Cover bowl with plastic and return to freezer to firm up, at least 30 minutes and up to 1 day.
Divide ice cream among 6 sundae dishes and top each portion with chilled strawberry sauce.
