For the tomatoes: Preheat the oven to 400 degrees F.
Lay the tomato quarters out on a baking sheet and sprinkle generously with salt. Drizzle with olive oil. Roast until golden brown around the edges, about 15 minutes. Set aside until serving.
Meanwhile, for the bacon and black pudding: Add the bacon to a large skillet and turn the heat to medium. Cook on both sides until the fat is rendered and the bacon is crispy, 3 to 4 minutes per side. Set aside on paper towel-lined plate to drain. Add the black pudding links to the skillet and cook, turning as needed, until heated through and brown on all sides, about 7 minutes. Add to the bacon and reserve until serving. Reserve the skillet with the fat.
For the mushrooms: Heat the skillet with the fat over medium heat. Add the shallot and a pinch of salt and cook, stirring, until translucent, about 3 minutes. Add the mushrooms in one layer and cook, undisturbed, until browned, about 3 minutes. Turn the mushrooms and add the butter to the skillet. Cook, stirring, until soft, about 3 minutes. Add the chives and toss to combine. Reserve until serving.
For the poached eggs: Fill a large, wide pot halfway with water and bring to a boil. Add the vinegar and reduce the heat until the water in the pot has no more bubbles. Very close to the surface of the water, crack the eggs and gently drop each egg into the water. Cook until the whites are set but the yolks are still runny, 3 to 4 minutes. Remove the eggs with a slotted spoon. (If making ahead of time, transfer the poached eggs to a bowl of cold water to store. When ready to serve, reheat the poaching liquid and warm the eggs through, taking care not to overcook them.)
To serve: Divide the tomato slices, bacon, black pudding, mushrooms, eggs and toast between two plates. Enjoy!