Begin by preheating the oven to 400 degrees Fahrenheit. Carefully slice off the top third of the garlic head to expose the cloves. Place the bulb on a piece of aluminum foil, approximately 12 by 12 inches in size. Drizzle with olive oil and season with a pinch of salt and several grinds of black pepper. Wrap the foil around the garlic, ensuring there is space between the foil and the exposed cloves, then seal the edges tightly. Bake the garlic until the cloves are deep golden brown and tender, which should take about 45 to 60 minutes. Allow to cool slightly, then squeeze the roasted cloves from their skins.

Meanwhile, in a large pot, add the potatoes and bay leaf, filling with enough cold water to cover the ingredients by an inch. Generously season with salt. Bring the mixture to a boil over medium-high heat, then reduce the heat to medium-low and simmer until the potatoes are fork-tender, approximately 12 minutes. Drain the potatoes well, discarding the bay leaf, and set the empty pot aside.

Next, pass the drained potatoes and roasted garlic through a potato ricer or food mill back into the reserved pot, ensuring they are finely mashed and smooth. Place the pot over medium-low heat, and incorporate the warmed heavy cream and butter using a wooden spoon until the mixture is creamy and well-combined. Gradually add half of the Gruyere and mozzarella, stirring vigorously until the cheese is fully melted and incorporated. Add the remaining Gruyere and mozzarella, continuing to stir until all the cheese has melted and the potatoes are smooth and stretchy, resembling the texture of melted mozzarella.

Finally, taste the mixture and adjust the seasoning with salt and pepper as necessary. Transfer the Aligot to a serving bowl, garnish with chopped chives, and serve immediately for the best experience.