Set a Dutch oven or large pot over medium heat and add the olive oil. Add the ground chicken, season with salt and pepper, and cook, breaking it up as you go, until mostly cooked through and crumbly, about 5 minutes. Transfer the chicken to a bowl and drain any excess liquid from the pot.

Add the butter to the pot. Place the lemon halves cut-side down and cook until deeply golden, about 3–4 minutes.

Lower the heat to medium-low and add the garlic slices and red pepper flakes to the butter while the lemons sear, letting them gently sizzle until fragrant. Remove the lemon halves and set aside.

Add the gnocchi, cooked chicken, and chicken stock to the pot. Stir to combine and bring to a simmer.

Cook for about 5 minutes, stirring every minute or so, until the gnocchi is tender and the liquid becomes slightly creamy.

Stir in the grated cheese, then add the kale and cook just until wilted, about 1 minute. Taste and season with additional salt and pepper as needed.

Squeeze one lemon half directly into the pot to finish. Slice the remaining lemon half for serving.

Spoon into bowls, top with extra cheese, and finish with the caramelized lemon wedges.