Preheat oven to 400ºF 

Trim carrots, saving stems for salsa verde. Toss carrots with olive oil, salt, pepper, cumin and cayenne. Roast in preheated oven for 15–20 minutes until caramelized and tender. Place roasted carrots in a food processor and pulse until smooth. Add the butter, sour cream and chicken stock and blend until smooth. Season with salt and pepper. Heat about an inch of oil in a heavy-bottomed skillet. “Soften” tortillas by holding them over a flame and searing on both sides, or simply by microwaving. Fill with 2–3 tablespoons of the carrot purée. Roll tightly and secure with a toothpick. Fry the flautas in heated oil until crispy on both sides and drain on a paper-towel-lined plate. 

In a mortar and pestle or food processor, mash garlic to a paste, add in carrot tops, mashing again until roughly chopped. Add the cilantro and mint, and mash greens until fine. Add lime zest and lime juice, streaming in olive oil, then stir to combine. Season with chile powder, salt and pepper. Just before serving, stir most of the crumbled cheese into the salsa. 

Arrange the flautas on a serving dish with a drizzle of the salsa verde, a sprinkle of the remaining Cotija cheese, and perhaps a few slices of lime.  The salsa verde is to serve on the side, for people to dip their flautas.