What Bea likes is the fact that I am eating spicy food; it's not me that she likes, it's something I do. That thing - eating spicy food - is characteristic of me. Other things that are characteristic of me - my hair colour, my voice, my limp ... - get a 'my'.
That said, many people do say 'me' and it is perfectly acceptable. I have to admit though that - while feeling it can sometimes sound a bit precious to say 'my <verb>ing' - I'm not entirely comfortable about saying me'. So I tend to work around the issue, by paraphrasing somehow.