Cut the butter into 8 pieces. Next to the stove place a medium course strainer over a bowl. This bowl is for the finished curd.
Place kumquats and the lemon juice in the bowl of a food processor. Process until the kumquats are puréed, about 1 minute. You may need to stop the processor after 30 seconds and scrape down the sides. Add the sugar and process for about 10 seconds.
Place the pureed kumquats in a 3-quart saucepan. Add the eggs 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.
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.
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.