How to count number of occurrences of each key in a MongoDB collection efficiently (one pass?)?

I'm a MongoDB noob and I have a MongoDB collection with millions of documents, each with tens of keys (average ~ 60, large variance) from a set of around 100. I'd like to get a count of each key across the entire collection. For example, if the documents in the collection were:

{"_id":0, "foo": 0, "bar":1, "baz":2}
{"_id":1, "foo": 0, "baz":7, "qux":11, "quux":13}
{"_id":2, "foo": 1, "bar":1, "quux":3}

then the desired output would be:

{"_id":3, "foo":3, "bar":2, "baz":2, "qux":1, "quux": 2}

I can "explode" the collection with $objectToArray, $unwind, $group, and then $count, but it's slow. Is there something that could do this efficiently in one pass through the collection? Something like,

[notional psuedocode]
output={}
foreach document:
  foreach key:
    if output.key exists:
      output.key+=1
    else:
      output.key=1

=> output: {key1: key1_count, ...}