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Heat olive oil in a wide, heavy skillet. Add the onions, 1/8 tsp salt and a few grinds of pepper. Cook, stirring often, over medium-heigh heat, until onions are browned and sticky/crispy, about 15 minutes. Remove pan from heat.
Meanwhile, in a second wide, heavy skillet, melt butter. Add all the mushrooms, garlic, 1/8 tsp salt and a few grinds pepper. Cook over medium-high heat, stirring frequently, until mushrooms are richly browned, about 15 minutes.
Increase heat to high, pour in cognac and cook until liquid disappears, 1 minutes. Remove pan from heat.
Split buns; lay them open-faced on a broiler rack. Cover both halves with cheese. Slide under the broiler just until cheese has melted, 1 to 2 minutes. Set buns on serving platter.
Heap mushrooms onto each bottom bun. Top with onions. Cover with top bun. Enjoy.
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