  We establish here a Quantitative Central Limit Theorem (in Wasserstein
distance) for the Euler-Poincar\'{e} Characteristic of excursion sets of random
spherical eigenfunctions in dimension 2. Our proof is based upon a
decomposition of the Euler-Poincar\'{e} Characteristic into different
Wiener-chaos components: we prove that its asymptotic behaviour is dominated by
a single term, corresponding to the chaotic component of order two. As a
consequence, we show how the asymptotic dependence on the threshold level $u$
is fully degenerate, i.e., the Euler-Poincar\'{e} Characteristic converges to a
single random variable times a deterministic function of the threshold. This
deterministic function has a zero at the origin, where the variance is thus
asymptotically of smaller order. Our results can be written as an asymptotic
second-order Gaussian Kinematic Formula for the excursion sets of Gaussian
spherical harmonics.
