Set the property of Domain to .mydomain.example
in each Cookies of two subdomains websites. Like:
Response.Cookies["test"].Value = "some value";
Response.Cookies["test"].Domain = ".mysite.example";
In Site A:
HttpCookie hc = new HttpCookie("strName", "value");
hc.Domain = ".mydomain.example"; // must start with "."
hc.Expires = DateTime.Now.AddMonths(3);
HttpContext.Current.Response.Cookies.Add(hc);
In Site B:
HttpContext.Current.Request.Cookies["strName"].Value