In a 2-quart pot set over medium-low heat, combine the cranberries, sugar, orange juice, and orange peel.
Continue heating, stirring occasionally, until the cranberries begin to burst. This process should take at least 15 minutes.
While heating the cranberries, in a large mixing bowl, whisk together the eggs and egg yolks.
Set a fine-mesh strainer over a large bowl. Once most of the cranberries have burst and the mixture has been gently simmering for several minutes, begin straining the mixture a little at a time. Use a wooden spoon or rubber spatula to press the cranberries firmly against the strainer, pushing as much liquid through as possible. Discard the remaining cranberry skins, and repeat until the entire mixture has been fully strained.
Transfer the strained cranberry juice mixture into a pitcher or measuring cup with a spout. While whisking the eggs from step 3 vigorously, very slowly pour the hot cranberry juice into the eggs, continuing to whisk until the mixture is fully combined and smooth.
Pour the entire mixture back into the pot and return it to low heat, stirring constantly.
Just as the cranberry curd is about to reach a boil, remove the pot from the heat and immediately whisk in the unsalted butter.
Once the butter has fully melted and been incorporated, strain the cranberry curd through the fine-mesh strainer directly into your prepared pie shell.
Transfer the pie to the refrigerator for at least 3 hours, or until the cranberry curd has fully set.