Here's another one. a little strange, since the quadrature rule involve the value of line integrals, but I guess you can write those using a separate quadrature rule...

<cite authors="Bojanov, Borislav; Petrova, Guergana">_Bojanov, Borislav; Petrova, Guergana_, [**Numerical integration over a disc. A new Gaussian quadrature formula**](https://doi.org/10.1007/s002110050358), Numer. Math. 80, No. 1, 39–59 (1998). [ZBL0911.65015](https://zbmath.org/?q=an:0911.65015).</cite>

>We construct a quadrature formula for integration on the unit disc which is based on line integrals over $n$ distinct chords in the disc and integrates exactly all polynomials in two variables of total degree $2n - 1$.

Cheers.