In a large pot, place the potatoes, garlic, thyme, peppercorns, and salt. Fill the pot with water, just to cover the potatoes and aromatics. Bring the water to a boil, then reduce heat to a simmer. Continue to simmer the potatoes until they are cooked all the way through, and easily pierced with a knife, about 20–30 minutes. Once cooked, drain the water and discard the thyme, garlic, and peppercorns. Cool the potatoes in baking pans. Lightly smash the potatoes by pressing down lightly with heel of your hand. They should remain mostly intact. Fry the precooked potatoes in a 350°F fryer, or a pot of oil heated to 350°F, until golden brown and crispy, about 5–7 minutes. After frying, season the potatoes with salt.
Blend all ingredients in a high-speed blender until very smooth and completely mixed, gradually increasing speed, about 2 minutes. There should not be any speckles. The mixture will be bright red. Transfer the purée to a squeeze bottle and store in the refrigerator until needed.