In a Dutch oven or deep sided skillet, heat 1 tablespoon of the butter over medium high heat. Add shallots, celery, thyme, and season with 1/2 teaspoon of salt. Cook, stirring occasionally, until onions begin to brown, less than 5 minutes. Stir in the vinegar, 1/4 cup of water, and the sugar and continue to cook until almost all the liquid has evaporated, about 5 minutes more.
Stir in the tomatoes and increase the heat to high; cook another 3 minutes. Stir in the lentils and another cup of water. Bring to a simmer, then reduce heat to medium and cook to soften the lentils, about 10 minutes.
Stir in the mustard; cook 5 minutes. Season with salt and pepper and stir in half the parsley. Remove from heat and cover to keep warm.
Melt the remaining 2 tablespoons of butter in a large skillet over medium heat. Break each egg, individually, into a small bowl, then slide into the skillet to fry until the whites are set, 3 to 5 minutes.
Meanwhile, divide lentils among serving plates, top with a fried egg, then garnish with remaining parsley. Serve with pita bread if desired.