如图，过$E$作$E M / / B C$，交$A D$于$N$，交$A B$于$M$，连接$M F$.

当$C$、$F$、$M$三点共线时，$E F + C F$的值最小.

∵$A C = 4$，$A E = 2$，∴$E C = A E = 2$，∴$A M = B M = 2$，

∴$A M = A E$，∵$A D$是$B C$边上的中线，$\triangle A B C$是等边三角形，

∴$A D \perp B C$，∵$E M / / B C$，∴$A D \perp E M$，

∵$A M = A E$，∴$E$、$M$关于$A D$对称.

∵$\triangle A B C$是等边三角形，

∴$\angle A C B = 60 ^ {\circ}$，$A C = B C$，∵$A M = B M$，

∴$\angle E C F = \frac {1} {2} \angle A C B = 30 ^ {\circ}$，故选选项C.