This is a partial duplicate to this MO question, I apologize for that. I'm asking since the answers there still do not allow me to work out an answer to my question, which is a bit more specific.

Informally speaking, an $E_\infty$-algebra is an algebra which is associative and commutative up to coherent homotopies, a notion that can be properly formalized via the language of operads. However, in characteristic zero, one can can obtain an equivalent category to $E_\infty$ algebras by taking commutative differential graded algebras and inverting the quasi-isomorphisms. This is the same mechanism by which one can define the category of $L_\infty$ algebras in characteristic zero by starting with differential graded Lie algebras and inverting quasi-isomorphisms.

However, in the $L_\infty$-case it is well known that $L_\infty$-morphisms between differential graded Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ can be also given an "explicit" definition as sequences of multilinear maps $\varphi_k\colon \wedge^k \mathfrak{g}\to \mathfrak{h}[1-k]$ which satisfy a set of quadratic equations. This explicit definition obscures the theory behind it but has nice "practical" consequences. For instance one immediately sees that an $L_\infty$-morphism between two Lie algebras (seen as differential graded Lie algebras concentrated in degree zero) is precisely the same thing as a Lie algebra morphism between them, or that an $L_\infty$-morphism from a Lie algebra $\mathfrak{g}$ to a differential graded Lie algebra $\mathfrak{h}$ concentrated in nonegative degree is the same thing as a morphism of differential graded Lie algebras from $\mathfrak{g}$ to $\mathfrak{h}$. Or even, if $\mathfrak{g}$ is a Lie algebra and $\mathfrak{h}=(\mathfrak{h}^{-1}\to \mathfrak{h}^0)$ is a differential graded Lie algebra concentrated in degrees -1 and 0, one has a complete, simple and explicit control on what an $L_\infty$ morphism from $\mathfrak{g}$ to $\mathfrak{h}$ would be.

I'm wondering whether something similar can be said for $E_\infty$-morphisms between differential garded commutative algebras.

For instance: can it be given an explicit description of the $E_\infty$-morphisms between a commutative algebra $A$ (seen as a differential graded commutative algebra concentrated in degree zero) and the differential graded commutative algebra $\mathbb{K}[1]\oplus \mathbb{K}$, where $\mathbb{K}$ is the base field? (the differential on $\mathbb{K}[1]\oplus \mathbb{K}$ is the zero differential and the multiplication $(\mathbb{K}[1]\oplus \mathbb{K})^0\otimes (\mathbb{K}[1]\oplus \mathbb{K})^{-1}\to (\mathbb{K}[1]\oplus \mathbb{K})^{-1}$ is the multiplication in $\mathbb{K}$).

I guess such an $E_\infty$-moprhism should reduce to the datum of two $\mathbb{K}$-linear maps $\varphi_0\colon A\to \mathbb{K}$ and $\varphi_1\colon A\otimes A\to \mathbb{K}$ satisfying suitable equations which shuld precisely encode the fact that $\varphi_0\colon A\to \mathbb{K}$ is a morphism of commutative algebras up to a homotopy given by $\varphi_1$. But I'm not able to find a rigorous statement along these lines anywhere in the literature.