Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?

Question

Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear that $$ \mathbb{F}_p \otimes_{HH^{\cdot}(\mathbb{F}_p)} \mathbb{F}_p$$ has homotopy groups $\mathbb{F}_p$ in cohomological degree $0$ and $1$. However I would like to know the algebra structure on it. It has an algebra structure because $HH^{\cdot}(\mathbb{F}_p)$ has an $E_2$ ring structure. Now from https://arxiv.org/pdf/1207.3461.pdf it seems that having homotopy groups as above leaves open many possibilities for what the algebra could be. For example it could be $Hom_{\mathbb{F}_p[[t]]}(\mathbb{F}_p,\mathbb{F}_p)$ (which I consider unlikely) or $Hom_{A}(\mathbb{F}_p,\mathbb{F}_p)$ for $A$ a totally ramified extensions of $\mathbb{Z}_p$ (giving a different algebra for each $A$). However, I am not sure how I can tell which algebra structure is on it.