In a heavy-bottomed stew pan, melt the butter over low heat.
Using a metal spoon, remove the milk solids from the surface -- they will appear as a creamy colored foam. This is essentially a quick way to clarify the butter. (The milk solids are the part of the butter that has a low burning point, so removing them helps prevent burning the contents of the pot.)
Turn the heat to medium-high and add all of the onions. The caramelization process will take about 1-½ to 2 hours. During the first 15 minutes, stir the onions every few minutes (I like to use metal tongs at this early point).
Once the onions are soft, reduce the heat to medium-low and stir periodically. Every so often the bottom and sides of the pan will begin to brown. This is what you want! Use a wooden deglazing spatula to work that back into the onions. When the onions are done, they will be very soft, golden brown, and very sweet.