Café Tortoni at Av. de Mayo 825 is well written about on the web, they have absolutely amazing chocolate. They serve churros too, but frankly, the churros are not as good as they are in México and are cold and more salty than sweet. On the back of the menu they claim churro is a national pastry - but this was the only place in Buenos Aires that I found serving it.
The chocolate drink is very delicious, thick, feels like it is just melted chocolate with some magical transformation so that there is no watery taste, but is still flowing liquid. This comes with a small pot of hot milk, and of course, three packets of sugar - everywhere in Buenos Aires, sugar is always available. In this case, skip the sugar, the chocolate is great just as is, and if needed, dilute it with the hot milk. This is definitely something not to be missed if visiting Buenos Aires. And then alternate between dunking the churro in the chocolate, and drinking the chocolate. This is good living!
While the cholocate is excellent, this place also has a lot of history, it has been open since 1858, visit the Café Tortoni web site for more info. The downside is that it is now on every tourists itinerary so there is a line to get in in the evening but it was never full in mid- to late-afternoon.