While the crust is baking, clean the food processor and set aside.
Add the cranberries, sugar, water, and salt to a medium saucepan and bring to a boil over medium-high heat. Cook for 10 minutes, stirring occasionally until the berries have burst.
Transfer the berry mixture to the food processor (or blender) and pulse for about 1 minute until smooth.
Pour the mixture through a fine-mesh strainer over the saucepan and use a spoon or rubber spatula to press the mixture through the strainer until only seeds remain. Remember to scrape off the bottom of the strainer. Discard the seeds.
In a medium bowl, whisk together the eggs and egg yolks and then add 1/3 of the cranberry mixture to the eggs slowly while continuing to whisk to temper the eggs.
Add the tempered egg mixture into the rest of the cranberry mixture in the saucepan, whisking constantly.
Cook the mixture over low heat for another 10 to 12 minutes, stirring constantly with a rubber spatula, making sure to scrape down the sides and bottom of the pan. The mixture is done when it has thickened and reaches 170°F and coats the back of a spoon.
Remove from heat and stir in the butter 1 tablespoon at a time, then stir in the lemon juice.
Pour the filling into the prepared pie crust and allow the pie to cool at room temperature for 1 hour before transferring to the refrigerator for an additional 4 hours.
When you're ready to serve, top with whipped cream and sugared cranberries.