That was a large part of the subject of my 1964 PhD thesis. While the motivation came from algebraic topology, the relevant algebra was published separately in the paper http://www.math.uchicago.edu/~may/PAPERS/3.pdf. It is very obvious from the case of abelian restricted Lie algebras with zero restriction what the minimal size of a $V(L)$-projective resolution of the ground field $k$ can be, where $L$ is a restricted Lie algebra with enveloping algebra $V(L)$. Section 6 of that paper constructs a resolution $X(L)$ of that minimal size for any $L$, using the theory of twisted tensor products.
The serious part of the mathematics is to construct a coproduct on $X(L)$ suitable for computing $Ext_{V(L)}^*(k,k)$ as an algebra. In characteristic $2$, the coproduct is coassociative and very explicit, as explained in Remarks 10, op cit, and $X(L)$ embeds into the bar constuction via a map of differential coalgebras. In char $p>2$, the price one pays for minimal size is that the coproduct is defined inductively and is not coassociative. The embedding into the bar construction is then only compatible with coproducts up to homotopy. The paper also gives a spectral sequence for computing the restricted Lie algebra cohomology from the cohomology of the underlying Lie algebra; that eases calculations and minimizes the difficulty caused by the cited non-coassociativity.