In a bowl, combine artichokes, 2 tablespoons oil, vinegar, parsley, and pepper flakes. Season with salt and set aside.
Bring milk and 3 cups water to a boil in a saucepan. Gradually add polenta and cook, stirring, until liquid is absorbed and polenta is tender, 18 to 20 minutes. Remove from heat and stir in remaining oil and Parmesan. Season with salt and pepper and cover. (If polenta becomes too thick, whisk in water by the tablespoon.)
Fill a high-sided skillet with 2 inches water; bring to a boil, then reduce to a simmer. Crack one egg at a time into a teacup or small bowl and gently slide into simmering water. Repeat with remaining eggs. Cook until white is just set and yolk is still loose, 3 to 4 minutes. Remove eggs from water using a slotted spoon.
Spoon polenta into serving bowls. Top with egg and sprinkle with artichokes.
Use polenta for a delicious alternative to breakfast porridge; you can also swap in quinoa.