食い切り and　食べ切り　probably mean the same thing, "a small portion", "a light meal", etc.

Small amount of food that can be eaten in on bite. Little portion.

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Baby burgers, teeny panini, little lobster rolls -- one-bite food is a trend, says Robin Uler, senior vice president for food and beverage for the 2,700 Marriott International properties. Expect to see the smaller bites on lounge menus, at catered affairs, as appetizers. The "small plates" craze will also continue. "We've finally learned how to share," Uler says.
http://www.washingtonpost.com/wp-dyn/content/article/2006/01...
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The Spanish word tapas doesn’t often show up on a Japanese restaurant menu, but One Bite isn’t like any other restaurant. This tiny restaurant isn’t a teppen-yaki steakhouse or a sushi joint. Its owner, Yokahama-born Erika Koike, has created her own version of an izakaya, which she likens to a Japanese tavern where patrons gather for drinks and a little bite of something. She quickly discovered that her customers preferred lots of bigger bites, too, so the menu ranges from starters to combo dinners that include salad and rice.
http://www.pitch.com/locations/one-bite-japanese-grill-29434...

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Note added at 5 hrs (2009-06-30 08:18:25 GMT)
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Also, it depends on where the restaurant was. If he worked abroad, it could be very well be the restaurants I mentioned above. In Japan, restaurants like 喰切り　江ぐち serve kaiseki, http://74.125.153.132/search?q=cache:UvlKk16F9F8J:homepage2....



　懐石料理は季節感が楽しめるが、コースの流れの縛りが強く、個々の料理となると印象の薄いきらいがある。三月に開店したばかりの「江ぐち」はそうした懐石のあり方に、一石を投じている。懐石をベースとしながらも、一皿一皿に強いアクセントを置く「喰切り」を看板とするのである。なにしろどのコースも、柱は四季を通じお椀代わりに織り込まれるすっぽん（丸なべ）。てんで破格なのだ。
Then it could be a traditional Japanese restaurant that mainly serves kaiseki full course ( course of several one-bite plates)

The comments above have led me to arrive at this translation.