Optimizing Bingo Caller Method
It is considered bad practice to use goto-statements in modern programming because it quickly gets hard to follow the structure, so avoid that. Instead you could use a while-loop in a kind of this manner:

while (list.Count &lt; 75)
{
  var number = random.Next(75);
  if (list.Contains(number))
    continue;
    ...
}




var item = random.Next(75);


This statement sets item to a number in the interval of \$[0, 75[\$, but you were supposed to handle the interval \$[1, 75]\$



instead of if (item &gt;= 61) it is more readable to write if (item &gt; 60)



It seems that GetNumber() is missing a stop-condition. It will run forever between goto Start and Start: when the list is full. (That may be your "time out"-problem).



Instead of the long list of if-statements, you could use the simple relation between the prefix's position in the word "BINGO" and the random number to extract it as:

string bingo = "BINGO"; // Make it a class field.
int index = ...
char prefix = bingo[index]


(it has something to do with 15).



The pattern of filling a list with found numbers tends to be slower and slower in response time as the size of the list grows because the next random number is more and more likely to be in the list. Potentially a valid number may never be found...

A better approach could be to create the list of the 75 numbers in the constructor in advance and then remove them one by one from the list when found by random.Next():

...
var number = list[random.Next(0, list.Count)];
list.Remove(number);
...


This will decrease in response time and you avoid the looping. 



A solution could be: 

  public class BingoCaller
  {
    private Random random;
    List&lt;int&gt; list;
    string bingo = "BINGO";

    public BingoCaller(Random random)
    {
      this.random = random;
      list = Enumerable.Range(1, 75).ToList();
    }

    public string GetNumber()
    {
      if (list.Count == 0)
        return "";

      var number = list[random.Next(0, list.Count)];
      list.Remove(number);
      char prefix = bingo[(number - 1) / 15];
      return $"{prefix}{number}";
    }
  }