1. The problem asks to find $\cot \theta$ given that $\sec \theta = 2 \times \frac{\sqrt{3}}{3}$.\n\n2. Recall the identity relating $\sec \theta$ and $\cos \theta$: $$\sec \theta = \frac{1}{\cos \theta}.$$\n\n3. Calculate $\sec \theta$: $$\sec \theta = 2 \times \frac{\sqrt{3}}{3} = \frac{2\sqrt{3}}{3}.$$\n\n4. Find $\cos \theta$ by taking the reciprocal of $\sec \theta$: $$\cos \theta = \frac{1}{\sec \theta} = \frac{1}{\frac{2\sqrt{3}}{3}} = \frac{3}{2\sqrt{3}}.$$\n\n5. Simplify $\cos \theta$ by rationalizing the denominator: $$\cos \theta = \frac{3}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{2 \times 3} = \frac{\sqrt{3}}{2}.$$\n\n6. Use the Pythagorean identity to find $\sin \theta$: $$\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{\sqrt{3}}{2}\right)^2 = 1 - \frac{3}{4} = \frac{1}{4}.$$\n\n7. Therefore, $$\sin \theta = \frac{1}{2}$$ (taking the positive root assuming $\theta$ is in the first quadrant).\n\n8. Recall the definition of $\cot \theta$: $$\cot \theta = \frac{\cos \theta}{\sin \theta}.$$\n\n9. Substitute the values: $$\cot \theta = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3}.$$\n\nFinal answer: $$\cot \theta = \sqrt{3}.$$