Blanch haricots verts in boiling water 2 minutes; drain and rinse with cool water.
Mix eggs, sugar, cornstarch, and salt in a bowl with chopsticks or a fork. Strain through a fine-mesh strainer.
Heat an 8-in. nonstick skillet over medium heat. When hot, very lightly oil with an oiled paper towel (if you have too much oil, the egg will not form a pancake). Pour 1 tbsp. egg mixture into pan, swirling it around the bottom to form a thin, even omelet with no browning (lower heat if it starts to brown). Cook 1 minute, until surface of omelet is dry; no need to flip. Carefully lift out onto a plate, trying to avoid folds and wrinkles. Repeat with remaining egg mixture, layering each onto the one below, until you've made 10 omelets. Roll omelets and slice into 1/8-in.-thick ribbons of egg.
Whisk broth, sesame paste, ground sesame seeds, miso, soy sauce, rice vinegar, sesame oil, and sugar together in a bowl.
Boil udon (see "Udon Essentials," below). Rinse well in a large bowl of ice water.
Arrange udon on a platter. Top with alternating piles of haricots verts, bell pepper, cucumber, carrot, and egg ribbons, with pickled ginger in center. Serve with a pitcher of dressing on the side, or pour the dressing over the salad.
Udon Essentials Udon (wheat-flour noodles): Store-bought fresh-frozen noodles have a supple texture that's closest to homemade, while the dried ones tend to be thin and flabby. To cook store-bought fresh-frozen udon, drop the frozen block into boiling water. When the water boils again, drain. Cook udon right before serving; the noodles get sticky as they sit.
Make ahead: Broth, up to 2 hours at room temperature. Vegetables and egg, up to 1 day, chilled.
Note: Nutritional analysis is per serving with 3 tbsp. dressing.