Peel carrots; remove tops and trim. With a vegetable brush, scrub carrots; rinse. In a covered medium saucepan, cook carrots in a small amount of boiling, lightly salted water for 7 to 9 minutes or until tender; drain and cool slightly. Cut each carrot in half lengthwise. Transfer carrots to a large bowl; sprinkle carrots with salt and pepper.
In a small bowl, whisk together vinegar, honey and mustard. Slowly add oil in a thin, steady stream (vinaigrette will thicken as oil is added), while continuing to whisk. Drizzle vinaigrette over carrots; toss to coat. Set carrots aside.
Cut off mushroom stems even with caps; discard stems. Using a spoon, remove gills underneath the mushroom caps. Lightly rinse mushroom caps. Wipe mushrooms with a clean, damp cloth or paper towel. Gently pat dry with paper towels. Sprinkle both sides of mushrooms with salt and pepper.
Place flour in a shallow dish. Place eggs in a second shallow dish. Place panko in a third shallow dish. Coat mushrooms with flour. Dip mushrooms in eggs; dip in panko to coat all sides.
In a large skillet, heat olive oil over medium heat. Carefully add mushrooms to hot oil. Cook for 4 to 8 minutes or until mushrooms are tender and cooked through, turning once halfway through cooking. Transfer to a cutting board. Cut warm mushrooms in half.
To serve, on each salad plate, place two mushroom halves. Sprinkle with goat cheese. Remove carrots from vinaigrette. Arrange 8 carrot halves in the center of the mushrooms. Sprinkle with chives.