Let $k$ be a field (if necessary for my question, we can assume its characteristic to be zero). In the non derived context, we can then define deformations of an associative algebra $S$ as follows:

If $A \in \mathfrak{A}rt_k$ is an Artin ring, a deformation of $S$, is a pair $(\mathcal{S},\pi_A)$, where $\mathcal{S}$ is an $A$-algebra and $\pi_A: \mathcal{S}\otimes_A k\to S$ is an isomorphism of $k$-algebras.

Two such deformations $(\mathcal{S},\pi_A)$ and $(\mathcal{S}',\pi'_A)$ are said to be isomorphic, if there exists an isomorphism of $A$-algebras $\varphi: \mathcal{S}\to\mathcal{S}'$, such that $\pi'_A\circ (\varphi\otimes_A id_k)=\pi_A$.

Then the functor

$Def_S: \mathfrak{A}rt_k \to Set\;;\; A \mapsto \{deformations\; over\; A\}\;/\;isomorphisms$

is a classical (underived) deformation functor.

Now my question is, how exactly can we extend this deformation functor, into (a model of) the derived setting,if we allow for more general $E_n$ or $E_\infty$ deformations?

Personally, I'm most familiar with Manettis approach to the derived situation, but every explicit extension would be welcome.