Whisk the lime juice, fish sauce, water, sugar, garlic, and jalapeño together in a bowl until the sugar dissolves. Transfer a small portion of the dressing to a separate large bowl and set it aside for the greens. Stir the carrots into the remaining dressing and let them sit to soften slightly.

Heat the oil in a large nonstick skillet over high heat until it just begins to smoke. Add the asparagus and cook, stirring occasionally, until the spears are browned in spots but still crisp-tender, about 4 minutes. Transfer the asparagus to a plate and place it in the freezer for 5 minutes to halt the cooking and lock in the color.

Toss the watercress with the reserved tablespoon of dressing in the large bowl and divide it among salad plates. Combine the chilled asparagus and mint with the carrot mixture, then arrange it over the watercress. Sprinkle with peanuts and serve immediately.