For the beans: Add the beans, ketchup, maple syrup, Worcestershire, mustard, bacon and 2 teaspoons salt to a small pot, bring to a simmer over medium heat and cook for 10 minutes. Remove from the heat and keep warm.
Heat an indoor grill pan over medium-high heat. Brush lightly with canola oil. Arrange the mushrooms and tomatoes on the pan and sprinkle with salt. Grill until they begin to brown, about 2 minutes. Flip and cook until just heated through. Remove from the heat and set aside.
Add the sausage links to the pan. Cook, turning as needed, until crispy and browned on all sides and cooked through in the center, about 7 minutes. Set aside and keep warm.
Grill the brioche until golden brown, about 2 minutes per side.
Add enough canola oil to coat the bottom of a large nonstick skillet and heat over medium. Crack the eggs into the skillet and adjust the heat to low. Sprinkle the eggs with salt and pepper. Cover the skillet and cook until the whites are firm but the yolks are still runny, about 2 minutes.
Arrange the grilled tomatoes and mushrooms over the cooked whites of the eggs. Slide the eggs off onto a large plate. Top with the arugula leaves, sprinkle with most of the chives and drizzle with the extra-virgin olive oil.
Serve the black pudding and toast on a side dish and the beans in a small cast-enamel pot. Sprinkle the beans with the remaining chives.