Question: I had a friend who said there is a slight difference in the two phrases. But I don't see any difference between the two expressions. Could anyone explain it? 

Thanks,
Rachel

Answer: 
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> 最后吃的菜 = 最后吃的(that you eat last) + 菜(dish/food);
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> so, 最后吃的菜 = "the dish that you eat last"
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"最后吃的菜" is a noun phrase and it is "菜" essentially.

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> 吃最后的菜 = 吃(eat) + 最后的菜(the last dish / the leftovers)；
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> so, 吃最后的菜 = “eat the last dish / leftovers”
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"吃最后的菜" can be regared as an imperative sentence and it means "eat something". If you put a subject at the beginning of it（"我吃了最后的菜"/“我要吃最后的菜”，etc.）, it would be easier to understand.
Let's say you went to a nice restaurant for a nice meal. 

Case 1: you had the meal, course by course. In this case 最后吃的菜=(吃)最后的菜=dessert

Case 2：you left in the meal for some reason. In this case 最后吃的菜=the dish you didn't finished. (吃)最后的菜=dessert.