Warm 1 tablespoon olive oil in a 10-inch skillet over medium-high heat. Add Swiss chard stems only and 1 large minced shallot. Cook, stirring often, for 3 to 4 minutes, until tender.
Add 3 minced garlic cloves, 1/2 teaspoon cumin, 1/2 teaspoon salt, 1/2 teaspoon paprika, 1/4 teaspoon red pepper flakes, and 1/4 teaspoon turmeric and stir for 1 more minute, until very fragrant. Stir in finely chopped Swiss chard leaves, 1/4 cup thinly sliced basil, and 1/4 cup water or heavy cream and cook for 2 to 3 minutes, until leaves are tender and most of the liquid has evaporated.
Reduce heat to medium-low. Using the back of a wooden spoon, create four 2-inch divots in the chard mixture. Crack an egg into each divot. Season eggs with a pinch of salt. Cover and cook until whites are almost set, 4 to 6 minutes.
Sprinkle 1/2 cup crumbled feta cheese over top, then cover, and cook until the whites are set and the yolks are cooked to your liking, 2 to 4 minutes more.
Serve into shallow bowls and garnish with remaining 1/4 cup thinly sliced basil.