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Heat oil in large skillet on medium-high heat. Add shallot; cook and stir about 5 minutes or until tender. Add spinach, vinegar, 1/4 teaspoon of the thyme and salt; cook and stir about 1 minute or until spinach is wilted. Remove spinach mixture from pan. Set aside.
Meanwhile, break eggs to separate the yolks and whites, adding all the egg whites into large bowl while keeping each yolk in separate small bowls or leaving each yolk inside the shell; set aside. Beat egg whites and remaining 1/4 teaspoon thyme with wire whisk until soft peaks form. (This should take about 5 minutes.)
Spoon egg white mixture into 4 mounds in same skillet on medium-high heat. Make an indentation in the center of each egg white mound using the back of a spoon. Sprinkle spinach mixture over egg whites. Place an egg yolk into each indentation. Cover skillet. Cook 3 to 5 minutes or until egg yolks are to desired consistency.