You've heard of the "clean plate club." Some of us grow up with moralizing parents who instill an eating ethic of wasting no food. But there's more to this than just cultural programming. The "clean plate" syndrome, at least in part, has to do with how our minds work, with our minds' reliance on the notion of a category.
Let me explain: when we want something, we want a something rather than a certain amount of that something. When we want a banana, we want its taste (substance), rather than its size or shape (form). At the same time, it's hard for our minds to envision the banana substance abstracted from its form. So, when the mind wants banana substance, it ends up wanting it in the form of a banana. Thus, the thought "I want banana (substance)" unconsciously morphs into the thought "I want a (one whole) banana."
As a result, a desire for a taste, mediated by category-driven perception, predetermines the serving size. We see things in units - and we end up eating them in units - forgetting that these naturally-occurring units have nothing to do with our physiological needs. In sum, the mind perceives and consumes in categories. And since a category is a unitary concept, eating half a banana leaves us with a sense of being unfinished, with a lack of closure. Therefore, we clean our plates to unconsciously prevent a sense of lacking closure. It would be too odd to think that you had a banana when you know you only had half of it. A half of a banana isn't a banana, after all. So, our decision to stop eating, instead of relying on fullness, becomes a matter of whether we are done eating a particular category of food (like a banana).
To help you guide your decision to stop eating by a consideration of substance, not form, experiment with re-thinking your portion size decisions. Instead of thinking to yourself that you want an apple (which means a whole apple, a unit of apple fruit), rephrase your desire in terms of the flavor you want. More specifically, try to say to yourself that what you want is not an apple per se, but a taste of an apple. This way, when having experienced the taste of the apple, you would feel finished, even if you are not finished with the rest of the apple.