Fasting from food is unique to us humans. Other animals may go without eating for any number of natural reasons. A domesticated animal may even be trained to refuse food as a conditioned behavior. But an animal has no spiritual nature; it does not have the faculty of volition, of free will, and so cannot deliberately choose to abstain from what its instincts, appetites, or conditioning demand. It cannot fast. Neither can angels fast; having no physical bodies, they have no need of food in the first place. To engage in fasting, one must be both physical and spiritual at the same time. We humans are the only creatures that fit the bill. It's a privilege, when you think of it; we can offer to our Creator something which no other creature can offer.
And so, the kind of fast that the discipline of Lent prescribes is the kind that engages both our physical and our spiritual natures. A non-religious person may fast solely for reasons of health or weight control. A hedonist may fast to heighten his pleasure in eating afterward. Such fasts could not be considered true Lenten practices, because they are merely physical; they do not connect with the spiritual partner of fasting which is prayer.
Likewise, a fast that does not connect with its other partner, almsgiving or mercy, is not a true Lenten fast. The religious person may fast severely and pray earnestly, but if he fails to give in concrete ways, his fast is incomplete, and his spiritual discipline is pointless. Chapter 58 of Isaiah rails against this kind of false fast, and Jesus condemned those who prayed and fasted rigorously, but whose hearts were far from both God and their neighbor in need. (e.g. Lk.18:9-14, Lk.5:29-35)
Fasting, then, as both a spiritual and physical reality is a sort of bridge uniting prayer and mercy. The essential harmony of these elements is perhaps best summarized by St. Peter Chrysologus:
Fasting is the soul of prayer, mercy is the lifeblood of fasting. So if you pray, fast; if you fast, show mercy; if you want your petition to be heard, hear the petition of others. If you do not close your ear to others, you open God's ear to yourself
- Sermo 43: PL 52, 320. 322.
So, hold the mayo. And hold the burger, too. Hold the pickle, bun, tomato... lettuce... pray.