Hi iammann,

I have created sample. Refer below code.

Global.asax

<%@ Import Namespace="System.Web.Configuration" %>
protected void Application_PreRequestHandlerExecute(Object sender, EventArgs e)
{
    // only apply session cookie persistence to requests requiring session information
    if (Context.Handler is IRequiresSessionState || Context.Handler is IReadOnlySessionState)
    {
        var sessionState = ConfigurationManager.GetSection("system.web/sessionState") as SessionStateSection;
        var cookieName = sessionState != null && !string.IsNullOrEmpty(sessionState.CookieName) ? sessionState.CookieName : "ASP.NET_SessionId";

        var timeout = sessionState != null ? sessionState.Timeout : TimeSpan.FromMinutes(20);

        // Ensure ASP.NET Session Cookies are accessible throughout the subdomains.
        if (Request.Cookies[cookieName] != null && Session != null && Session.SessionID != null)
        {
            Response.Cookies[cookieName].Value = Session.SessionID;
            Response.Cookies[cookieName].Path = Request.ApplicationPath;
            Response.Cookies[cookieName].Expires = DateTime.Now.Add(timeout);
        }
    }
}

Web.config

<configuration>
  <system.web>
    <compilation debug="true" targetFramework="4.0"/>
    <sessionState cookieName="Product"></sessionState>
  </system.web>
</configuration>

C#

protected void Page_Load(object sender, EventArgs e)
{
    string cookieName = (ConfigurationManager.GetSection("system.web/sessionState") as SessionStateSection).CookieName;
    string cookiePath = Request.Cookies[cookieName].Path;
    DateTime cookieExpiry = Request.Cookies[cookieName].Expires;
    string cookieValue = Request.Cookies[cookieName].Value;
    Response.Write(string.Format("Path: {0} <br/>Expiry: {1}<br/>Value: {2}", cookiePath, cookieExpiry, cookieValue));
}

VB.Net

Protected Sub Page_Load(sender As Object, e As EventArgs) Handles Me.Load
    Dim cookieName As String = TryCast(ConfigurationManager.GetSection("system.web/sessionState"), SessionStateSection).CookieName
    Dim cookiePath As String = Request.Cookies(cookieName).Path
    Dim cookieExpiry As DateTime = Request.Cookies(cookieName).Expires
    Dim cookieValue As String = Request.Cookies(cookieName).Value
    Response.Write(String.Format("Path: {0} <br/>Expiry: {1}<br/>Value: {2}", cookiePath, cookieExpiry, cookieValue))
End Sub

Screenshot

I have created the sample by refering the below link.

Setting Path and Expiration for session cookie in asp.net