How can I delete current session in Chrome?

I don't know if this is what you were looking for but anyway, a similar functionality is built right into chrome:

Right-click -> Inspect element.

Go to resources tab and enable tracking if you haven't already.

Now scroll down the left hand sidebar thingy until you see cookies, under it click your domain name and to delete a cookie just right-click on its name and delete

For clarity: You want to be able to ONLY delete the cookies of a specific site OR even just single cookies of a given site, right?

There's a few extensions that let you do stuff like that.

If you just want to delete all cookies for the site opened in the current tab, check out Remove Cookies For Site.

If you want more options, like editing, deleting only certain cookies of a site, Edit This Cookie might be your extension. You'll get a menu for all cookies of the current site:

You can use the following procedure in Chrome:

1. Hit F12 or open the contextual menu (right click), then "Inspect element"
2. Go to the Application tab
3. On the right sidebar menu, go to Application > Clear storage
4. Uncheck everything but Local and session storage
5. Scroll to the very down and click Clear site data

Hope it helps