1. **State the problem:** We need to find the magnitude $|z|$ of the complex number $$z = \frac{(1 + i)^5}{(\sqrt{3} - i)^2}.$$\n\n2. **Recall the formula for magnitude:** For any complex number $w = a + bi$, its magnitude is $|w| = \sqrt{a^2 + b^2}$. Also, the magnitude of a quotient is the quotient of magnitudes: $$|\frac{w_1}{w_2}| = \frac{|w_1|}{|w_2|}.$$\n\n3. **Calculate magnitudes of numerator and denominator:**\n- Numerator: $1 + i$ has magnitude $$|1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}.$$\n- Denominator: $\sqrt{3} - i$ has magnitude $$|\sqrt{3} - i| = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2.$$\n\n4. **Apply powers to magnitudes:**\n- Numerator magnitude raised to 5: $$|1 + i|^5 = (\sqrt{2})^5 = (2^{1/2})^5 = 2^{5/2} = 2^{2 + 1/2} = 2^2 \times 2^{1/2} = 4\sqrt{2}.$$\n- Denominator magnitude raised to 2: $$|\sqrt{3} - i|^2 = 2^2 = 4.$$\n\n5. **Calculate $|z|$:**\n$$|z| = \frac{|(1 + i)^5|}{|(\sqrt{3} - i)^2|} = \frac{4\sqrt{2}}{4} = \sqrt{2}.$$\n\n**Final answer:** $$|z| = \sqrt{2}.$$