Trim beet greens, leaving about 1 inch of stem, and rinse off any dirt. Place whole beets in a large pot, cover with water, and bring to a boil. Reduce heat and simmer 30–45 minutes, or until a knife slides into the center with little resistance. Let cool slightly, then slip off the skins under cool running water. Slice or cube evenly.

If you boiled the beets, strain and reserve at least 1 cup of the beet cooking liquid. If you roasted the beets or are using canned beets, measure 1 cup of beet juice from the can, or combine beet juice with low-sodium vegetable broth to reach 1 cup.

In a small bowl, whisk together 2 tablespoons cornstarch with 1/4 cup of the beet liquid until completely smooth. Set aside near the stove.

In a medium non-reactive saucepan, add the remaining beet liquid (about 3/4 cup), apple cider vinegar, sugar, salt, pepper, and if using, orange zest and a pinch of cloves or allspice. Stir well and bring to a gentle simmer over medium heat, stirring occasionally until the sugar is fully dissolved.

Stir the cornstarch slurry (it settles quickly) and slowly whisk it into the simmering sauce. Continue whisking until the sauce thickens and turns glossy, 2–4 minutes. It should lightly coat the back of a spoon.

Gently stir in the sliced or cubed beets, turning them to coat well in the sauce. Bring back to a very gentle simmer and cook 5–10 minutes, stirring occasionally, just until the beets are warmed through and nicely glazed without breaking apart.

Remove the pan from the heat and stir in the butter until melted and the sauce looks silky. Taste and adjust seasoning with additional salt, sugar, or vinegar as needed.

Serve the Harvard beets warm as a side dish, or let them cool to room temperature. The sauce will thicken slightly as it cools, and the flavors deepen after an hour or two. They can also be chilled and served cold.