If making a kidney-friendly version, place the potatoes in a large pot and cover with double the amount of water. Bring to a boil and cook, 10 to 15 minutes. Drain the potatoes in a colander, and discard the water.
In a saucepan over medium low heat, add the fat or butter.
Add the shallots and garlic and cook, stirring occasionally, until very soft and golden, 15 to 20 minutes.
Add the potatoes, milk, thyme and salt, and stir to combine.
Bring to a simmer, skimming occasionally, until potatoes are fork tender, 20 to 25 minutes.
Remove the liquid from the pot, reserving at least ¾ cup of the liquid. The remaining liquid can be drizzled over the potatoes when serving, or used in other recipes if desired.
Using a potato masher or a wooden spoon, mash the potatoes. Add the reserved cooking liquid, until the mashed potatoes achieve the desired consistency. Adjust seasoning to taste.
Serve the potatoes, drizzled with additional cooking liquid and garnished with thyme leaves.