Roast and sauté peppers:
Roast bell peppers on racks of gas burners over high heat, turning with tongs, until skins are charred, 6 to 8 minutes. (Alternatively, broil peppers on rack of a broiler pan about 5 inches from heat, turning occasionally, 15 to 25 minutes.) Transfer to a bowl and let stand, tightly covered, until cool. Peel peppers, discarding stems and seeds, and finely chop.
Heat 1 1/2 tablespoons oil in a 10-inch heavy skillet over moderately high heat until hot but not smoking, then sauté peppers, stirring frequently, until tender and excess liquid is evaporated, 5 to 10 minutes. Stir in basil and salt and pepper to taste. Transfer to a bowl and cool.
Cook chard leaves in a 4- to 6-quart pot of boiling salted water , uncovered, until tender, 2 to 3 minutes. Drain in a colander and rinse under cold water to stop cooking. Squeeze handfuls of chard to remove excess moisture, then finely chop.
Cook shallot in remaining 1 1/2 tablespoons oil in cleaned skillet over moderate heat, stirring occasionally, until softened and beginning to brown, 4 to 5 minutes. Add chard and cook, stirring occasionally, until mixture looks dry, 2 to 3 minutes. Stir in salt and pepper to taste and cool.
Prepare egg mixtures and bake omelet:
Preheat oven to 450°F.
Break 4 eggs into each of 2 bowls, then add 1/8 teaspoon salt and pepper to taste to each and whisk to combine eggs in each bowl. Whisk 3 tablespoons crème fraîche into 1 bowl of eggs until smooth, then stir in bell pepper mixture. Whisk 1/4 cup crème fraîche into other bowl of eggs until smooth, then stir in chard mixture.
Pour bell pepper eggs into oiled loaf pan and bake in a hot water bath in middle of oven until firm to the touch, 18 to 20 minutes (about 13 minutes if using round cake pan).
Pour chard eggs into loaf pan and continue to bake until layer is firm, 18 to 20 minutes more (about 13 minutes if using round cake pan).
While chard layer bakes, break remaining 5 eggs into a bowl, then add remaining 1/8 teaspoon salt and pepper to taste and whisk to combine. Whisk in Cheddar and remaining 1/4 cup crème fraîche, then pour cheese eggs into loaf pan and bake until layer is lightly browned and slightly puffed, about 20 minutes (about 16 minutes if using round cake pan). Transfer loaf pan to a rack and cool omelet 5 minutes.
Invert a long platter over loaf pan and invert omelet onto platter. Serve hot, warm, or at room temperature