Prepare prebaked tart shell for custard-filled tart or pie.
Broil pepper about 4 inches from heat, turning as needed, until blackened on all sides, 15-20 minutes. Let steam in closed plastic bag or covered bowl 5 minutes to loosen skin.
Peel off blackened skin, then cut out stem. Cut pepper in half, remove seeds and white pith, and cut into 1/2-inch strips.
Heat oven to 400 degrees.
Heat butter in small skillet over medium heat. Add shallot and cook, stirring occasionally, until softened, about 3 minutes. Let cool.
Whisk remaining ingredients except whole chives together in mixing bowl, then whisk in cooled shallot. Place tart pan on baking sheet. Pour egg mixture evenly into tart shell. Arrange pepper strips in sunburst pattern on tart and place whole chives between strips.
Bake tart until custard is set and pastry is lightly browned, 15-20 minutes. Transfer tart to wire rack and let cool 10 minutes before serving.