Checking area of overlap between two dataframes with parallel dask GeoPandas

I have two different GeoDataFrames: One of which contain polygon squares in a large grid. The other contains larger, and fewer, polygons. I wish to calculate the area of overlap within each of the grid squares with the other, larger squares.

To do so, I made a simple loop method

for _, patch in tqdm(layer.iterrows(), total=layer.shape[0], desc=name):
    # Index of intersecting squares
    idx = joined.intersects(patch.geometry)
    intersection_polygon = joined[idx].intersection(patch.geometry)
    area_of_intersection = intersection_polygon.area
    joined.loc[idx, "value"] += area_of_intersection

In an attempt to speed up this method, I converted the layer DataFrame, which contains the larger patches to a Dask-DataFrame.

I implemented it the following way:

def multi_area(patch, joined=None):
    # Index of intersecting squares
    idx = joined.intersects(patch.geometry)
    intersection_polygon = joined[idx].intersection(patch.geometry)
    area_of_intersection = intersection_polygon.area
    joined.loc[idx, "value"] += area_of_intersection
    return joined["value"]

layer_dask = dask_geopandas.from_geopandas(layer, npartitions=8)

with ProgressBar():
    joined["value"] = layer_dask.apply(multi_area, meta=joined, joined=joined, axis=1).compute(scheduler='multiprocessing')

This, however, returns the error AttributeError: 'GeoDataFrame' object has no attribute 'name', and at this point I am unsure if this is the optimal way of doing it, and what I am doing wrong.

The job I will be doing will have 400 million grid squares, so I am planning on batching this calculation out on smaller areas later, as I can't come up with a smarter way of doing it...