Place the peeled and cut potatoes in a large pot or saucepan. Cover with cold water by two inches and add two teaspoons of salt.
Bring to a simmer over medium-high heat and then reduce heat to medium low. Continue to cook until the potatoes are really tender and falling apart, about 20-25 minutes. Go ahead and fish one out and test it with a knife to make sure it's done.
Drain the potatoes in a strainer and give them a shake or two to remove as much water as possible.
Meanwhile, melt the butter and warm the milk or half-and-half in the microwave. The milk just needs to be warm to the touch, not boiling.
Pass the cooked potatoes through a potato ricer or food mill and transfer them back into the pot. Stir the melted butter into the riced potatoes until thoroughly coated. Add in ½ cup of milk or half-and-half and stir to combine. If you would like the potatoes to be a little looser in texture, add some or all of the remaining milk.
Season to taste with salt and pepper. Set the pot back on the stove over low heat and stir constantly until the mashed potatoes are hot. Serve immediately topped with additional butter if desired.