Let $x$ and $y$ be positive real numbers such that
\[\frac{1}{x + 2} + \frac{1}{y + 2} = \frac{1}{3}.\]Find the minimum value of $x + 2y.$
By the Cauchy-Schwarz inequality,
\[((x + 2) + 2(y + 2)) \left( \frac{1}{x + 2} + \frac{1}{y + 2} \right) \ge (1 + \sqrt{2})^2.\]Then
\[x + 2 + 2y + 4 \ge 3 (1 + \sqrt{2})^2 = 9 + 6 \sqrt{2},\]so $x + 2y \ge 3 + 6 \sqrt{2}.$

Equality occurs when $(x + 2)^2 = 2(y + 2)^2,$ or $x + 2 = (y + 2) \sqrt{2}.$  Substituting into $\frac{1}{x + 2} + \frac{1}{y + 2} = \frac{1}{3},$ we get
\[\frac{1}{(y + 2) \sqrt{2}} + \frac{1}{y + 2} = \frac{1}{3}.\]Solving, we find $y = \frac{2 + 3 \sqrt{2}}{2}.$  Then $x = 1 + 3 \sqrt{2}.$

Hence, the minimum value we seek is $\boxed{3 + 6 \sqrt{2}}.$