For the Irish brown bread: Position a rack in the middle of the oven and preheat the oven to 400 degrees F. Lightly dust a baking sheet with all-purpose flour; set aside.
Whisk together the all-purpose and whole-wheat flours, baking soda and salt in a large bowl to combine. Add the buttermilk, brown sugar and 4 tablespoons of the melted butter and mix with your hands until the dough holds together, about 1 minute.
Lightly flour a clean work surface and turn out the dough. Knead until it forms a fairly smooth ball. Work the dough into a flat round about 7 inches in diameter and 2 inches thick. Place on the prepared baking sheet and, using a sharp knife, slice an "X" across the top, edge to edge and about 1/2 inch deep. Brush the top with the remaining 2 tablespoons butter and sprinkle the oats evenly over the top.
Bake until the internal temperature registers 190 to 200 degrees F on an instant-read thermometer and the bread makes a hollow sound when tapped, 35 to 40 minutes. Transfer to a wire rack and cool completely before slicing, at least 2 hours and up to 8 hours.
For the tomato jam: Heat the canola oil in a medium saucepan over medium heat. Add the tomatoes and granulated sugar and cook, stirring occasionally, until the tomatoes are soft and starting to break down, about 20 minutes.
Add the red wine vinegar, salt, coriander, black pepper, mustard seeds, cloves and chile flakes. Cook, stirring frequently, until sticky and jammy, about 15 minutes. Stir in the honey and taste--the jam should be sweet but tangy. Add more honey if desired. Set aside; keep warm.
For the sandwiches: Slice the brown bread into 8 1/2-inch-thick slices and toast (you might have to cut the slices in half to fit them in the toaster). Butter the bread with about 2 tablespoons of the butter and keep warm.
Heat the oil in a medium skillet over medium-high heat until it begins to shimmer. Add the sausages cut-side down and cook until golden brown on all sides, 1 to 2 minutes per side. Set aside and keep warm.
Heat the remaining 2 tablespoons butter in a large nonstick skillet over medium heat. Crack the eggs one-by-one into a small bowl and slip them into the hot butter. Cook until the whites are barely set, 1 to 2 minutes, then scatter the Cheddar over the top, cover and cook until the Cheddar is melted and the whites are set but the yolks are still runny, 1 to 2 minutes more.
Spread the tomato jam on the toast, place a cheesy egg on top and top each egg with a few sausage halves. Garnish with chives and serve immediately.