Place the apricots in a large saucepan and add enough water to cover them.
Cook over medium heat for about 20 minutes, or until apricots are very soft. Drain the apricots, reserving the cooking liquid. Set the pan aside for later use.
Place the apricots in 2 batches in the bowl of your food processor fitted with the metal blade. Puree by pulsing on and off until the apricots are smooth. Add some of the reserved cooking liquid, if necessary.
Scrape the puree back into the large saucepan and stir in the brown sugar and lemon juice. Cook over low heat, stirring continually, until the sugar is dissolved, about 5-10 minutes.
Let cool, then store in an airtight container in the refrigerator, until needed.
You can use this as is, for a filling, or if the flavor is too concentrated, add it to some buttercream. Then you will have an apricot buttercream filling.
I torte my cakes into four cake layers and like to do three filling layers as follows:
Cake, filling, cake, buttercream flavored with filling, cake, filling and final cake layer. By doing this, you will have two layers of the apricot filling, one on the bottom layer, and one on the top layer and a middle layer of the buttercream flavored with the filling. The buttercream will take on a very pale peachy color.
Just remember to pipe an icing dam around the layer before spreading the apricot filling.