Bring a medium pot of salted water to a gentle boil. Add the whole baby potatoes and simmer until just tender when pierced with a knife, about 12–15 minutes depending on size. Parboiling ensures the potatoes cook through quickly and gives a fluffy interior that will crush easily to create maximum surface area for crisping.

Preheat the oven to 400°F (200°C). While the potatoes cook, line a baking sheet with a clean kitchen towel or paper towel and drain the potatoes; let them steam-dry for 3–4 minutes. Dry potatoes crisp up much better when seared.

Using a heavy-bottomed skillet (10–12-inch/25–30 cm, oven-safe), heat 1 tbsp olive oil over medium-high heat. Add the potatoes in a single layer — you may need to work in two batches. Press each potato gently with a wide spatula or the bottom of a heavy glass to flatten to about 1/2-inch thickness. Flattening creates more contact with the pan so you get deeply browned, crunchy edges.

Cook the smashed potatoes undisturbed for 4–5 minutes until the underside is golden and crisp. Flip and cook the other side 2–3 minutes. Season with 1/2 tsp kosher salt and a generous grind of black pepper. Browning (the Maillard reaction) builds savory flavor and contrast to the tender inside.

Reduce heat to medium, push the potatoes to one side of the skillet and add the remaining 1 tbsp olive oil and the butter. When the butter foams, add the minced garlic, chopped thyme and smoked paprika; cook 30–45 seconds until fragrant — do this to bloom the spices and mellow raw garlic. Stir quickly to coat the potatoes in the aromatic oil-butter mixture.

Move the potatoes to create 3–4 small wells in the skillet (depending on size) and nestle the chopped kale around and between the potatoes. The kale will wilt quickly and absorb the flavored fat — this step integrates its bright, slightly bitter flavor into the dish.

Make shallow wells in the potatoes/kale mixture and crack two eggs gently into the wells (2 eggs per person). Season the egg whites lightly with salt and pepper. The wells help the eggs stay centered and distribute heat evenly while baking so the whites set without overcooking the yolks.

Scatter the lemon zest evenly over the pan and transfer the skillet to the preheated oven. Bake at 400°F (200°C) for 7–10 minutes depending on desired doneness: 7–8 minutes for runny yolks, 9–10 for jammy. Baking sets the whites gently while the residual heat keeps yolks creamy.

Remove the skillet from the oven and immediately sprinkle the grated Parmesan over the hot potatoes and eggs so the cheese melts slightly and adds a salty umami lift. Squeeze a lemon wedge over each serving to introduce bright acidity that cuts through the richness.

Serve the skillet family-style or transfer to plates: aim to scoop a mix of a crispy smashed potato, a bit of wilted kale and an egg (try to keep the yolk intact until you break it at the table). Finish with a crack of black pepper and more thyme if desired.

To reheat leftovers: gently warm in a 325°F (160°C) oven for 10–12 minutes covered with foil so the egg yolks remain soft and the potatoes re-crisp at the edges. Avoid microwaving whole to prevent rubbery eggs and uneven reheating.