using System;

using System.Linq;

using Newtonsoft.Json;

public class Program

{

    public static void Main()

    {

        var Lavori = new [] {

            new {

                WorkerName = "foo",

                datetime = new DateTime(2019, 01, 01)

            },

            new {

                WorkerName = "bar",

                datetime = new DateTime(2019, 02, 01)

            },

            new {

                WorkerName = "baz",

                datetime = new DateTime(2019, 03, 01)

            },

            new {

                WorkerName = "whatever",

                datetime = new DateTime(2019, 03, 01)

            }

        };

        var dictionary = Lavori

            .GroupBy(l => l.datetime)

            .ToDictionary(l => l.Key.ToString("dd.MM.yyyy"), l => l);

        string output = JsonConvert.SerializeObject(dictionary);

        Console.WriteLine(output);

    }

}