What you are looking for is DisplayFormatAttribute : 
public class MyModel 
{
    [DisplayFormat(DataFormatString = "{0:yyyy-MM-dd}", ApplyFormatInEditMode = true)]
    public DateTime DateChosen { get; set; }
    // Other properties goes there
}
And use your Html helper like this : 
 @Html.EditorFor(x=> x.DateChosen, new { htmlAttributes = new { @id = "choosetime" } })
Edit 1 :
Base on comment, if you need to access the string representation of the DateChosen property somewhere else in your code, you can update your model like that : 
public class MyModel 
{
    [DisplayFormat(DataFormatString = "{0:yyyy-MM-dd}", ApplyFormatInEditMode = true)]
    public DateTime DateChosen { get; set; }
    public string DateChosenString
    {
        return DateChosen.ToString("yyyy-MM-dd");
    }     
    // Other properties goes there
}