$(E G)_{Milnor} := colim \; G \ast G \ast \ldots \ast G,$

and canonically equips it with a continuous and free right action of $G$ that yields the structure of a CW-complex such that the action of $G$ permutes the cells. Consequently, the natural projection $(E G)_{Milnor} \to (E G)_{Milnor}/G$ is a model for the universal bundle $E G \to B G$ of locally trivial principal $G$-bundles over paracompact Hausdorff spaces?, or equivalently, of numerable bundle$G$-principal bundles over all Hausdorff topological spaces.

There is a generalisation of Milnor’s construction that works for topological groupoids, and reduces to the above case when the groupoid is $\mathbf{B}G$, the delooping of the group $G$.

John Milnor, Construction of Universal Bundles, I, Ann. of Math. 63:2 (1956) 272-284 jstor; Construction of Universal Bundles, II, Ann. of Math. 63:3 (1956) 430-436, jstor; reprinted in Collected Works of John Milnor, gBooks.