Heat broiler. Roast bell pepper on a broilerproof baking sheet, turning occasionally, until blackened in spots, 12-15 minutes. Place in a small bowl; cover tightly. Let sit 10 minutes. Peel and remove seeds.
Purée bell pepper, mango, brown sugar, vinegar, orange juice, lime juice, and cayenne in a blender, thinning with water as needed, until smooth; season with salt. Do ahead: Hot sauce can be made 1 week ahead. Cover and chill.
Whisk eggs and milk in a medium bowl. Melt butter in a large nonstick skillet over medium heat. Cook eggs, stirring, until mostly set but still runny in parts, about 5 minutes. Remove from heat and mix in cheese; season with salt and pepper. Build sandwiches with rolls, hot sauce, eggs, avocado, and cherry peppers.
