For the Garlic Mojito Sorbet: In a saucepan, combine the water, sugar, and garlic. Bring to a boil and cook for 5 minutes, until sugar is dissolved, but not at all colored. Remove from the heat, strain, reserving garlic, and cool completely. Mash the reserved garlic in a mortar and pestle and set aside.

In a blender, combine the cooled sugar water, lime, mint, and rum. Blend until well combined. Add the soda water and place in an ice cream machine. Process according to manufacturer's instructions.

For the Brittle: Line 2 baking sheets with parchment or waxed paper. Combine the water and sugar in a large nonstick heavy-bottomed saucepan over medium-high heat. Stir to dissolve the sugar, 4 to 5 minutes. Add the corn syrup, honey, and salt, and stir constantly and slowly with a wooden spoon until the mixture comes to a gentle boil, about 2 minutes. Continue stirring until the mixture reaches 290 degrees F on a candy thermometer. Remove the pan from the heat. Add the butter, reserved garlic puree, and baking soda and stir until the butter is completely melted, about 1 minute. Pour the mixture onto the baking sheets, being careful as it is very hot, and spread it evenly with a rubber spatula. Let cool completely.

Break the brittle into shards. Scoop the sorbet into decorative glasses. Place a shard of brittle in the sorbet and garnish with mint sprigs and/or lime.