Toast panko in a 12-inch skillet over medium-high, stirring constantly, until golden, 4 to 5 minutes. Set aside.

Cut carrots into thirds, and then cut each third in half lengthwise (or into fourths, depending on width of carrots).

Heat oil in a cast-iron skillet over medium until hot. Cook carrots, flat side down, in hot oil, without turning, until tender, 10 to 12 minutes. (The cut sides will be very dark.)

Remove skillet from heat; stir in ¼ cup water, honey, and sumac. Return to heat. Cook, stirring constantly, until most of the water has cooked off and carrots are coated, 4 to 5 minutes. Season with ½ teaspoon of the salt and ¼ teaspoon of the pepper, and transfer to an airtight container. Chill carrots about 30 minutes.

Meanwhile, pulse feta and cream cheese in a food processor until blended, about 5 pulses. Add whipping cream, lemon juice, and remaining ½ teaspoon salt and ¼ teaspoon pepper. Process mixture until smooth, 2 to 3 minutes. Set aside until ready to use.

Spread Feta Mousse onto bottom of serving plate, and top with carrots. Sprinkle carrots with toasted panko and mint before serving.