Preheat oven to 400°F. On a baking sheet, toss carrots with olive oil and season with salt. Bake until tender, about 45 minutes. Let cool, then slice into 1-inch pieces.
Meanwhile, place amaranth in a blender or food processor and pulse a few times until the amaranth cracks and acquires the texture of uncooked polenta. Heat a large pot over medium heat and add cracked amaranth. Toast lightly for 2 minutes, then add water and salt. Bring to a boil, reduce to a simmer, stir occasionally, and cook until thickened, about 20–25 minutes.
In a small saucepan, melt butter over medium heat. Add shallots and garlic and cook, stirring, until soft but not browned, about 5 minutes.
Place roasted carrots, shallots, and garlic in a blender. Add 1 cup of vegetable broth and puree until smooth. Add puree to cooked amaranth and stir together over medium heat. Continue to add in vegetable broth until polenta reaches a creamy texture. Continue to cook for 5 minutes more. Season with kosher salt to taste.
Garnish with herb oil and popped amaranth. To pop amaranth, heat a deep cast iron skillet on high for 5 minutes. Add 1/2 tablespoon of amaranth seeds and shake continously for about 30 seconds. Seeds will puff and turn white. Remove form heat quickly to avoid burning.