Heat a large cast-iron skillet over high heat. Add carrots and whole lime; cook, turning occasionally, until charred on all sides, about 3 minutes for lime and 7 minutes for carrots. Remove from skillet; let cool slightly.
Zest and juice lime into a large bowl; whisk in oil, mustard, agave, salt, black pepper, and coriander until smooth.
Shred carrots into bowl with lime mixture; stir in green onion, almonds, apricot, and serrano pepper, if using. Cover and refrigerate for at least 30 minutes before serving. Fold in basil just before serving.