I don't know of any automated way, no.
You could use something similar to this to make things faster:
- Draw circles, do 'Stroke to Path'
- Make a duplicate of the circles, Combine the duplicated circles
- Use the MultiBool extension set (
https://inkscape.org/en/~Moini/%E2%98%8 ... e-division) to cut all circles into pieces with Multiple Division, move combined path out of the way.
- Remove a few circle segments to make it look as if they were overlapping
- use 'select same' to get the all elements of a specific circle by color, do Path -> Union for each circle, fix stacking order
- Add a stroke, if you want it to look as it does in your image