Use the $expr operator which allows the use of aggregation expressions within the query language. This will give you the power to use the Date Aggregation Operators in your query as follows:
month = 11
db.mydatabase.mycollection.find({ 
    "$expr": { 
        "$eq": [ { "$month": "$date" }, month ] 
    } 
})
or 
day = 17
db.mydatabase.mycollection.find({ 
    "$expr": { 
        "$eq": [ { "$dayOfMonth": "$date" }, day ] 
    } 
})
You could also run an aggregate operation with the aggregate() function that takes in a $redact pipeline:
month = 11
db.mydatabase.mycollection.aggregate([
    {
        "$redact": {
            "$cond": [
                { "$eq": [ { "$month": "$date" }, month ] },
                "$$KEEP",
                "$$PRUNE"
            ]
        }
    }
])
For the other request
day = 17
db.mydatabase.mycollection.aggregate([
    {
        "$redact": {
            "$cond": [
                { "$eq": [ { "$dayOfMonth": "$date" }, day ] },
                "$$KEEP",
                "$$PRUNE"
            ]
        }
    }
])
Using OR
month = 11
day = 17
db.mydatabase.mycollection.aggregate([
    {
        "$redact": {
            "$cond": [
                { 
                    "$or": [ 
                        { "$eq": [ { "$month": "$date" }, month ] },
                        { "$eq": [ { "$dayOfMonth": "$date" }, day ] }
                    ] 
                },
                "$$KEEP",
                "$$PRUNE"
            ]
        }
    }
])
Using AND
var month = 11,
    day = 17;
db.collection.aggregate([
    {
        "$redact": {
            "$cond": [
                { 
                    "$and": [ 
                        { "$eq": [ { "$month": "$createdAt" }, month ] },
                        { "$eq": [ { "$dayOfMonth": "$createdAt" }, day ] }
                    ] 
                },
                "$$KEEP",
                "$$PRUNE"
            ]
        }
    }
])
The $redact operator incorporates the functionality of $project and $match pipeline and  will return all documents match the condition using $$KEEP and discard from the pipeline those that don't match using the $$PRUNE variable.