dg.differential geometry - Kähler structure on cotangent bundle?

This is true! I assume $M$ compact.

Method 1. Real algebraic geometry. Cf. this article. By a version of the Nash-Tognoli embedding theorem, one can realise $M$ as an real affine algebraic variety $V_mathbb{R}$, cut out by polynomials $f_i in mathbb{R}[x_1,dots,x_N]$. The complex variety $V_mathbb{C}$ will then be smooth in a small neighbourhood $U$ of $V_mathbb{R}$, hence Kaehler in that region, with $V_{mathbb{R}}$ as a Lagrangian submanifold. But $U$ is diffeomorphic to $T^ast M$. The resulting symplectic structure on $T^ast M$ may be non-standard; via the Lagrangian neighbourhood theorem, you can take the symplectic form to be the canonical one if you'll settle for a Kaehler structure only near the zero-section.

Method 2. Eliashberg's existence theorem for Stein structures. See Cieliebak-Eliashberg's unfinished book, Symplectic geometry of Stein manifolds, Theorem 9.5. We observe that $T^ast M$ has an almost complex structure $J$ (one compatible with the canonical 2-form, for instance) and a bounded-below, proper Morse function $phi$ whose critical points have at most the middle index (namely, the norm-squared plus a small multiple of a Morse function pulled back from $M$). In this situation Eliashberg, via an amazing chain of deformations, finds an integrable complex structure $I$ homotopic to $J$ such that $dd^c phi$ is non-degenerate. This makes $T^ast M$ Stein! His theorem only applies in dimensions $geq 6$ (this paper of Gompf explains what you have to check in dimension 4), so without doing those checks or appealing to other methods, the case of $M$ a surface is left out.

I think that the more precise version of Eliashberg's theorem, which may not yet be in the book, would tell us that the Stein structure is homotopic to an easy-to-write-down Weinstein structure on $T^ast M$ involving its canonical symplectic structure $omega_{can}$, hence that $dd^cphi$ is symplectomorphic to $omega_{can}$.