How to Make It
Preheat the oven to 375°. Cut off the top 1/2 inch of the head of garlic. Wrap the garlic in aluminum foil and bake for about 45 minutes, until very soft.
Meanwhile, in a medium saucepan, cover the beets with cold water and add a generous pinch of salt. Simmer the beets over moderately low heat until tender, 30 minutes. Drain and let cool slightly, then peel the beets.
On a work surface, using a mug or the bottom of a small bowl, gradually press down on the beets until they are about 3/4 inch thick and cracked around the edges; try to keep the beets whole.
In a large cast-iron skillet, melt the butter in the olive oil. Add the beets and cook over moderately high heat until crusty on the bottom, 4 minutes. Turn the beets and scatter the onion, thyme, rosemary and roasted head of garlic all around. Season with salt and pepper. Cook over moderately high heat, gently stirring the onion and herbs, until the onion is just soft and the beets are crusty, 4 minutes. Transfer the beets, onion and herbs to plates or a platter and scatter the arugula over the top. Squeeze the roasted garlic cloves from their skins and scatter them over the top. Spoon the pan juices over the arugula and serve.
The roasted garlic and boiled beets can be refrigerated separately for up to 2 days; let return to room temperature before proceeding.