MongoDB query on an indexed field takes forever to return a simple count

Question

I have a collection with over 1 billion documents. Some of them have the field extra.id but a majority don't. I have an index set as described below. When I run a simple query to get the count of documents with extra.id set, the query takes ages to complete. Actually, I have never waited long enough for it to complete. I'd expect <1s response and what I get instead is >15min (before I kill it). I am running MongoDB 3.4.7 on Ubuntu while connecting from a Windows client.

What is going on? I thought partial filter expression was going to help here because the query should simply return the count of indexed documents.

Query

db.handles.find({"extra.id":{$exists:true}}).count()

Index

{ 
    "v" : 1, 
    "name" : "extra.id_1", 
    "ns" : "pull.handles", 
    "background" : true, 
    "partialFilterExpression" : {
        "extra.id" : {
            "$exists" : true
        }
    }
}