My daughter's like this. I have to admit, it does drive me crazy. We're working on getting her to think with her stomach, not with her eyes.
Here's some stuff that helps.
If I think she's had enough, I tell her that she has to wait 5 minutes. If she's still hungry in 5 minutes, she can have something else. Sometimes it just takes our stomachs a little while to catch up.
If I'm having something to eat and she wants it, I tell her she can have one bite, but it's my breakfast/lunch whatever. Honestly, this is more about manners than it is about food. No one likes to be harassed into giving up their lunch!
If she's finished everything she's asked for, I check in with her to see if she's full, and then I remind her that she can't have something else to eat for 2 hours. If she leaves food behind, I tell her that's fine, but she can't have something else for 2 hours.
If she asks for something to eat and I think that she's not really hungry/isn't going to like it, etc the rule is that she has to eat at least half of it. So we used to have a big battle about yogurt - she was sure she liked it, but she'd always take one bite and decide she wasn't hungry. Finally I said if she asked for yogurt, that was all she got to eat until it was gone. It only took one time of having yogurt for snack and then the same one for lunch for her to realize that she didn't like it and to stop asking for it. Same thing with the "I want more/I'm full!" I DON'T believe in the clean plate club, but I also don't believe in wasting food and not thinking before you ask for more. It makes her more hesitant to ask for thirds if she knows she's really going to have to eat it all.
Finally, I often say to her "listen to your belly." Sometimes she'll decide that she only wants a "little bit" or will have more later.
Sorry this got so long. I do know the conflict between not wanting to starve a hungry child and having a child who learns that constant eating is not ok or healthy. Good luck.