@FrankScience not sure if this is what you're looking for, but here's a possible solution. let k be a perfect F_p-algebra, and let S_{W(k)} denote the spherical witt vectors. this is an E_oo-ring which is flat over the (p-complete) sphere spectrum S, and pi_0(S_{W(k)}) = W(k). consider the map L^2 S^3 -> BGL_1(S_{W(k)}) classifying the element 1+p in pi_0 GL_1(S_{W(k)}); the thom spectrum is the free E_2-S_{W(k)}-algebra F with a nullhomotopy of p.
by general formal properties, this is the same as the base-change to S_{W(k)} of the free E_2-algebra with a nullhomotopy of p, which by hopkins-mahowald is the base-change to S_{W(k)} of HF_p. Since S_{W(k)} is flat over S, this is just isomorphic to Hk, so k is a thom spectrum over S_{W(k)}
