Applying Henderson–Hasselbalch equation to amino acids: which pKa to use to calculate Z-/HZ ratio?
There are two approaches.


The two groups are treated as independent. You would calculate the ratio of protonated and deprotonated group one using the pKa for that group. Then, you would repeat that for the second group. Independent treatment would be a good approach, for example, to estimate the average charge of the hundreds of side chains in a typical protein that have a pKa near physiological pH.
The two pKa values are obtained experimentally from a titration.$$\ce{AH2 &lt;=&gt; AH- + H+ &lt;=&gt; A^2- + 2 H+}$$ Here, the $\ce{AH-}$ species could be ambiguous if the pKa values are close to each other (group 1 deprotonated and group 2 protonated or vice versa). Or, for a case like hydronium, water and hydroxide where the same atom can accept two protons, $\ce{AH-}$ would represent a single defined species. You can get the ratios of the concentrations from the pH and the respective pKa ($\ce{AH2}$ vs $\ce{AH-}$ via $K_1$, and $\ce{AH2}$ vs. $\ce{AH-}$ via $K_2$), and turn that into fractions of the total for each species. You can use that for a weighted average to calculate the average charge, for example.



  Let's say I want to know the ratio of HZ to Z− for a pH of 1.00 and 11.00, which pKa should be used in the Henderson–Hasselbalch equation and why?


This depends what HZ and Z- refer to. With two ionizable groups, there are really four species, $\ce{HZYH, HZY^-, ^-ZYH and ^-ZY-}$. If the pKa values are far apart, one of the mixed species would be very unlikely in aqueous solution (in your example, protonated carboxylate with deprotonated amino group). At pH 1.00, the amine would be 100% protonated, and the carboxylic acid would be mostly protonated (comparing its pKa of 2.33 to a pH of 1.00). At pH 11, the carboxylic acid would be 100% deprotonated, and the amine would be mostly deprotonated (comparing its pKa of 9.74 to a pH of 11.00).