Place the pureed kumquats in a 3-quart saucepan. Taste the mixture before adding the eggs. If it is too tart, add 1 or 2 tablespoons or additional sugar and whisk it in. But remember that the curd is supposed to be tart.
Add the whole eggs and egg yolks, and whisk to combine. Place pan over medium heat and whisk continuously until the mixture reaches 180 degrees, about 6-8 minutes. Remove the curd from the heat and whisk in all 8 tablespoons of butter. Pour the curd into the strainer and use a large spatula to press the curd through the strainer and into the bowl below. This step will take a while because the curd is thick and has a lot of solids (kumquat skin and seeds).
Cover the curd with plastic wrap by placing a piece of plastic wrap over the bowl, lightly press it onto the surface of the curd. Place the bowl in the refrigerator and allow to finish cooling.
Once the curd is cooled it can be used as a spread on toast and biscuits or used as a filling for a tart. This curd is also thick enough to use as a cake filling.
Storage
The curd can be stored in the refrigerator in an airtight container for up to two weeks. It can be frozen for up to one year in a freezer safe container. When ready to use the curd after freezing place it in the refrigerator the night before to thaw before using.
Notes
I use a medium coarse strainer to allow some of the kumquat skin pieces to come through. If you want a perfectly smooth curd use a fine mesh strainer.