1. Bring 1 1/2 cups water and the sugar to a boil in a medium saucepan, stirring to dissolve sugar. Remove syrup from heat.
2. Meanwhile, whirl ginger in a food processor to mince. Add 3/4 cup water and whirl about 1 minute to extract juice. Pour mixture into a fine strainer set over a bowl and press to extract liquid; discard ginger. Stir liquid into syrup, pour 1/2 cup of mixture into a small bowl, and pour the rest into a 9- by 13-in. pan. Let liquids cool. Chill the 1/2 cup ginger liquid, covered, until used.
3. Cover pan with foil and freeze until slushy at edges, about 2 hours. Stir well with a fork, breaking up any lumps, then freeze until solid, at least 3 hours.
4. Meanwhile, stir apricots into liquid in bowl, then chill at least 30 minutes and up to 2 hours, stirring occasionally.
5. Scrape frozen mixture with a fork to make large flakes. Spoon ginger ice into bowls and drizzle each with some sweetened condensed milk. Drain apricots, then spoon over desserts. Garnish with mint sprigs.
Make ahead: For ginger syrup, up to 1 day, chilled. For flaked ginger ice, up to 1 week, frozen.