Recursively apply a projection to a nested object in Mongo DB

I have a data structure like this in MongoDB. It can have many levels:

{
    _id: ObjectId("123"),
    name: "Top level",
    someData: "hello world",
    parent: null,
    children: {
        "456": {
            _id: ObjectId("456"),
            name: "Nested Obj",
            parent: "123",
            someData: "lorem ipsum",
            children: {
                "963": {
                    _id: ObjectId("963"),
                    name: "2nd level nesting",
                    parent: "456",
                    someData: "some data",
                    children: {}
                }
        },
        "798": {
            _id: ObjectId("798"),
            parent: "123",
            name: "Another entry",
            someData: "more data here",
            children: {},
        }
    }
}

Is there a way to make a query that would allow me to:

1. Recursively apply a projection to all nested objects.
2. Create a calculated field that contains the path to get to each object.

For example, say that I am only interested in the _id and name fields, I would expect to see this as a result of the query:

{
    _id: ObjectId("123"),
    name: "Top level",
    path: []
    children: {
        "456": {
            _id: ObjectId("456"),
            name: "Nested Obj",
            path: ["123"]
            children: {
                "963": {
                    _id: ObjectId("963"),
                    name: "2nd level nesting",
                    path: ["123", "456"]
                    children: {}
                }
        },
        "798": {
            _id: ObjectId("798"),
            name: "Another entry",
            path: ["123"]
            children: {},
        }
    }
}