Place beans in a large bowl and using a potato masher, mash beans. Add beef, breadcrumbs, egg, onion, garlic, jalapeño, Worcestershire, chili powder, cumin and salt. Using your hands, mix well to combine and shape into 6 patties. Place on large plate.
Preheat grill to medium heat and lightly spray with cooking spray. Grill burgers and peppers for about 12 minutes, turning occasionally until peppers are golden and burger is no longer pink inside. Sprinkle shredded cheddar evenly over top of each burger patty, then place cut Cheddar or mozzarella fangs to melt over patties.
Place burgers into buns and top with peppers to serve.
In a large bowl, combine milk with 1 tsp (5 mL) of the sugar until dissolved. Stir in yeast and let stand for about 5 minutes or until yeast begins to froth. Whisk in egg, butter, remaining sugar and squid ink. Stir in 2 1/4 cups (560 mL) of the flour and salt until ragged dough forms. Turn out onto floured surface and knead dough for about 5 minutes, adding more flour (if necessary) until soft, smooth dough forms.
Place dough in greased bowl; cover and let rise for about 1 hour or until doubled in size.
Gently deflate the dough and divide into 6 pieces. Shape each piece into a round ball about 3 inches (7.5 cm) across. Place buns on parchment paper-lined baking sheet; cover and let rise for about 1 hour or until doubled in size. Brush tops with egg white and sprinkle with sesame seeds.
Bake in preheated 375°F (190°C) oven for about 15 minutes or until bottom is golden and top is firm. Let cool completely before slicing to use.