What is a lagrangian such that Euler-Lagrange equation (not sure if it's correct form for this case)
$$\frac{\partial \mathcal{L}}{\partial g_{\mu\nu}}=\partial_\lambda\frac{\partial \mathcal{L}}{\partial (\partial_\lambda g_{\mu\nu})}.$$
Gives us Einstein field equations?
This is almost certainly answered elsewhere, but the Hilbert Action, from which Einstein's equation can be derived, is:
$$S = \int d^{4}x\;\left(\sqrt{|g|}\frac{1}{16\pi G}R + \mathcal{L}_{m}\right)$$
taking the variation is pretty complicated (there are second derivatives of the metric in the action, and you have to deal with gauge invariance) and best looked up in a textbook, though. But note, that by this definition, we define $T_{ab} = \frac{\delta \mathcal{L_m}}{\delta g^{ab}}$
With the cosmological constant $\Lambda$ included, the Hilbert action for empty space is $$S = \frac{c^4}{16 \pi G} \int (R-2 \Lambda) \sqrt{-g} \, \mathrm{d}^4 x. $$
Wikipedia calls it the Einstein-Hilbert action, but this is wrong. The action is due to Hilbert, not to Einstein.
