Pour rosé into a plastic container and freeze until almost solid at least 6 hours.
Meanwhile, bring sugar and ½ cup water to a boil in a medium saucepan. Cook, stirring constantly until sugar dissolves. Add strawberries and remove from heat and allow to sit for 30 minutes to infuse.
Strain through a fine-mesh sieve into a small bowl refrigerate until chilled.
Scrape rosé into a blender. Add lemon juice, ½ cup strawberry syrup and 1 cup of ice. Purée until smooth.
Transfer to freezer and freeze until frosé is thickened 25–35 minutes.
Blend again and spoon into glasses. Serve topped with prosecco and fresh lime.