Preheat the oven to 400°F (205°C). Roast the whole head of garlic to transform raw pungency into soft, sweet, spreadable cloves. Cut the top 1/4" off the garlic head to expose the cloves, drizzle 1 tsp olive oil over the cut surface, wrap tightly in foil and roast for 35–40 minutes until very soft and nutty. Roasting mellows harsh sulfur notes and gives a caramelized sweetness that becomes the backbone of the sauce.

While the garlic roasts, scrub the potatoes clean (leave skins on for texture and nutrients). Place them in a large pot, cover with cold water by 1 inch and add 1 tsp kosher salt. Start with cold water so the potatoes heat evenly and the centers cook through without the outsides falling apart. Bring to a gentle boil, then simmer until just tender when pierced with a skewer, 15–20 minutes depending on size. Drain well.

Line a rimmed baking sheet with parchment or lightly oil it. Arrange the drained potatoes in a single layer. Use the bottom of a sturdy glass, a potato masher, or a small skillet to press each potato until its flesh breaks and the skin fans out — roughly 1/4" thick. Smashing increases surface area so more edges will crisp in the oven; aim for even thickness for consistent browning.

Brush the smashed potatoes with 2 tbsp olive oil and sprinkle with 1 tsp kosher salt and 1/4 tsp black pepper. Roast on the middle rack at 425°F (220°C) for 20–25 minutes, then flip each potato and roast another 8–12 minutes until edges are deeply golden and crisp. High heat helps evaporate surface moisture quickly to produce the sought-after lacy crust.

While potatoes roast, make the garlic chips and brown butter. Thinly slice 2–3 cloves of raw garlic (or 3–4 thin slices from the roasted head if you prefer) into 1/16" slices. Heat 1 tbsp neutral oil in a small skillet over medium heat. Fry the garlic slices, stirring, until light golden — 45–75 seconds. Remove with a slotted spoon to a paper towel and season immediately with a pinch of salt. Frying at moderate heat yields crisp chips without burning — burnt garlic becomes bitter quickly.

In a small saucepan, melt the butter over medium heat. Swirl frequently and watch for it to foam, then develop light brown flecks and a nutty aroma — about 3–4 minutes. Immediately remove from heat to stop browning. Browning the butter (beurre noisette) concentrates flavor and adds a toasty, almost caramel element that pairs beautifully with roasted garlic and potatoes.

Squeeze the roasted garlic head into a bowl and mash into a paste with the back of a spoon or fork. Mix this roasted-garlic paste into the warm brown butter, along with the lemon zest, 1 tbsp lemon juice, thyme, and the remaining 1 tbsp olive oil. The warm butter helps emulsify the roasted garlic into the sauce; lemon brightens and cuts through richness, while thyme adds herbal lift.

When the potatoes are crisp and golden, transfer them to a serving platter. Drizzle the brown-butter-roasted-garlic sauce evenly over the potatoes, then sprinkle the grated Parmesan (if using), chopped parsley, and the reserved garlic chips. Toss gently or spoon the sauce over to coat; the contrast between crisp edges and the silky, aromatic butter is the hallmark of this dish.

Finish with an extra pinch of flaky sea salt and a grind of black pepper. Serve immediately while the potatoes are hot and the edges are at their crispiest. The finishing salt wakes up the flavors and the heat from the potatoes helps release the aroma of the butter-garlic sauce.