I remember exactly how I felt about a year ago when I saw somebody pull out an electric hand mixer and start whipping the crap out of a batch of boiled potato slices. I cringed, audibly gasped, and prepared myself for gluey, sticky potatoes.
How do you avoid gluey mashed potatoes? First, you have to understand why that happens in the first place. Potato cells are filled with starch granules surrounded by a wall of pectin. When these cells are handled too vigorously, whether by boiling or overzealous mashing, the pectin walls rupture or are sheared apart, spilling out starch which binds with water and turns into a gel or paste. To prevent this, you have to somehow both bind the starch and break the cells apart without rupturing them.
Fortunately, we can use the gelling property of starch to our advantage, as long as we keep the starch gel within the cell walls. The potato starch swells inside the cells when it is cooked in 160° water for around 30 minutes. The temperature is high enough for the starch to do its magic, but not hot enough to dissolve the pectin cell walls. Once the starch has swelled, you can cool the potato slices down either under a cold tap or in the refrigerator. This causes the starch gel to permanently solidify—the starch is now retrograded. Now that the starch is fixed, the potato can be handled more aggressively. The potato slices are then simmered for another 30 minutes at 190° in salted water, hot enough to dissolve the cell walls.
At this point, the potatoes can be pureed using your choice of instruments, however I prefer using a potato ricer. A ricer creates a uniformly smooth puree while damaging as few of the cells as possible. Since we retrograded the starch, the danger from cell damage is not as great, so you could use an electric mixer without creating glue, but I find that a ricer is just as easy to use.
Adding butter and milk is purely based on individual taste. I find that there is no point in adding butter unless you can taste it, and that usually ends up being a good guideline. Add enough cold butter until you can taste it. I only add enough milk or cream to modify the texture to my liking. If you salted your water enough during the second simmer, the potatoes should be seasoned properly.
After the puree tastes and feels good, I sometimes pass it through a tamis to ensure that there are no lumps. It is an extra step that is not absolutely necessary, but does make a difference in the final product.
The beauty of these potatoes is that they can be refrigerated and reheated with no deterioration in texture. Everybody has pulled out leftover mashed potatoes before only to find that they have become a potato brick, molded to the container. This technique results in fluffy potatoes at any temperature, even after refrigeration, so is useful for make-ahead dinners or pre-prep. During reheating, a splash of milk can sometimes help to refine the texture.