In non-stick skillet, whisk together curry paste, garlic and vegetable broth over medium high heat. Cook until liquid has reduced by half, about 3 minutes. Remove from heat and whisk in honey and olive oil until smooth.
Stir in lemons and carrots until they are completely coated and scrape into a single layer on a foil-lined rimmed baking sheet. Roast in the centre of the oven, stirring often, until the carrots are tender and caramelized, about 25 minutes.
Meanwhile, in a small bowl, stir together, yogurt with cilantro, mint, olive oil, salt and pepper.