Construct a Circuit computing all boolean functions over n bits Let $ n∈N $ . Construct a circuit with $ C_n(x_1,\dots,x_n) $ with $ 2^{2^n} $ outputs $ y_1,\dots,y_{2^{2^n}} $ which computes all
  distinct boolean functions $ f_i:\{0,1\}^n→\{0,1\}$ such that $ y_i $ is the
  output of $ f_i $ and satisfying $ Size(C_n)\in O(2^{2^n}) $


Remark: The circuit is in a model of fan-in 2.

I've already found a solution in $ O(2^{2^n + n}) $ but I can't find less even recursively.
Given the circuit $C_n$, you can compute $C_{n+1}$ using $O(2^{2^{n+1}})$ additional gates. Indeed, each output gate of $C_{n+1}$ is of the form
$$
(\lnot x_{n+1} \land y_i) \lor (x_{n+1} \land y_j)
$$
for some $i,j$, so can be computed from the outputs of $C_n$ using $O(1)$ additional gates.

In total, the circuit $C_n$ uses $O(2^{2^n} + 2^{2^{n-1}} + \cdots) = O(2^{2^n})$ gates.