For the eggs: Position an oven rack towards the bottom of the oven and preheat to 350 degrees F.
Heat a 10-inch saute or cast-iron pan over medium-low heat with the olive oil. Sweat the shallots until tender, about 8 minutes. Add the kale and saute until wilted, 2 to 3 minutes.
Add the green chiles, olives, chicken stock, anchovy paste and garlic and bring to a simmer. Braise, covered, until the kale is tender and the liquid has been almost fully absorbed, about 10 minutes.
Spread the mixture flat in the pan and make 4 wells to place the eggs. Lay a basil leaf in each well. Add an egg to each well on top of the basil leaves and sprinkle everything with salt and pepper. Sprinkle with the sun-dried tomatoes and feta.
Bake on the bottom rack until the eggs are set but still runny, about 10 minutes.
For the toast: Meanwhile, toast the bread in a toaster and rub with garlic cloves, then spread with the butter. Grate Pecorino over the eggs and sprinkle with chives, then serve with the toast.