Peel the yautía with a potato peeler or paring knife, removing the thin brown skin. Chop the yautía into roughly 1–2 inch pieces and rinse under cold water.
Place the rinsed yautía pieces in a pot and add enough water to cover them plus about 2 inches (5 cm). Add 1 tablespoon of the salt to the water.
Bring the pot to a boil over high heat, then reduce the heat and simmer until the yautía is fork-soft when pierced (about 20–25 minutes).
Drain the cooked yautía and discard the cooking water. Let the pieces sit in the colander for a minute to shed excess water.
Return the drained yautía to the pot or a large bowl. For the smoothest texture, press the yautía through a potato ricer or food mill. If you don’t have one, mash thoroughly with a potato masher and then beat briefly with a sturdy spoon or hand mixer until smooth.
Add the ½ stick salted butter to the hot mashed yautía and mix until melted and incorporated.
Gradually stir in the ¾ cup milk until the mash reaches a smooth, creamy consistency.
Stir in the 1 tablespoon garlic powder. Taste and add more salt if needed.
Transfer to a serving dish, top with the Dominican sauteed red onions if using, and serve warm.