Solution:

Since we have square roots, we can try to change the inside of each radical to be a perfect square.

Note that we can rewrite the expression as $$\sqrt{6 - 6\sqrt2 + 3} + \sqrt{6 + 6\sqrt2 + 3}.$$

Factoring and simplifying gives us $$\sqrt{(\sqrt6 - \sqrt3)^2} + \sqrt{(\sqrt6 + \sqrt3)^2}$$ $$= \sqrt6 - \sqrt3 + \sqrt6 + \sqrt3 = 2\sqrt6.$$

Thus, B is the correct answer.