[{"_id":"1","a":"b","b":"wanted value"},{"_id":"2","a":"c","c":"wanted value 2"}]

db.collection.aggregate([{"$addFields":{"arr":{"$objectToArray":"$$ROOT"}}},{"$addFields":{"v":{"$reduce":{"input":"$arr","initialValue":null,"in":{"$cond":{"if":{$eq:["$$this.k","a"]},"then":"$$this.v","else":"$$value"}}}}}},{"$set":{"arr":{"$filter":{"input":"$arr","as":"tuple","cond":{$eq:["$$tuple.k","$v"]}}}}},{"$set":{"result":{"$arrayToObject":"$arr"}}},{"$replaceRoot":{"newRoot":"$result"}}])