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Heat the butter and canola oil in a cast iron pan over high heat until the butter begins to sizzle. Add the onion and cook until soft and slightly caramelized.
While the onions are caramelizing, put the potatoes in a large pot of cold, salted water, bring to a boil and cook until nearly tender, but a knife inserted into the centers still meets a little resistance, 10 to 12 minutes, or cooked through. Drain well and smash with the back of a wooden spoon.
Add the potatoes to the onions packing it evenly with the back of a metal spatula to form a cake. Season with salt and pepper and cook until the underside is golden brown, about 10 minutes. Invert a plate or baking sheet without sides over the pan and carefully invert the potato cake onto it. Carefully slide the potato cake back into the skillet and cook until the bottom is just set and pale golden brown, about 4 minutes. Spread the cheese on top, cover the pan with a lid and let melt. Transfer the potato cake to a cutting board cheese-side up and cut into wedges. Garnish with chopped fresh chives.
For the roasted tomatoes with bread crumbs: Preheat the oven to 350 degrees F. Put the tomato halves in a lightly buttered baking dish, cut-side up, drizzle with olive oil , honey, and sprinkle with panko bread crumbs and paprika. Season the tomatoes with salt and pepper and roast for 20 minutes. Remove the tomatoes from the oven and sprinkle with parsley. Drizzle with olive oil.
For the sauteed mushrooms: Heat the canola oil into a saute pan and saute the mushrooms until soft. Add the fresh thyme leaves and season with salt and pepper.