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1
Heat 1 tablespoon oil in a large saucepan over medium heat until shimmering. Add onions, season with salt and pepper, and cook, stirring frequently, until lightly browned and a pale brown patina builds up on the bottom of the pan, about 10 minutes. Add 1 tablespoon water and scrape up browned bits. Repeat, cooking, adding water, and scraping up bits until onions are soft and very sweet, about 30 minutes total. Transfer to a bowl and set aside.
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2
While onions cook, heat remaining oil in a large skillet over high heat until shimmering. Add sliced mushrooms, season lightly with salt and pepper, and cook, stirring occasionally, until mushrooms have released all their liquid, the liquid has evaporated, and the mushrooms have deeply browned, about 15 minutes. Add half of garlic and cook, stirring constantly, until fragrant, about 30 seconds. Add soy sauce and half of lemon juice and stir to combine. Season to taste with salt and pepper. Transfer to a bowl and set aside.
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3
Combine mayonnaise, remaining lemon juice, tomatoes, and garlic in a food processor. Process until a rough puree is formed, about 30 seconds, scraping down sides as necessary.
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4
Preheat a panini press. Spread both sides of rolls with sun-dried tomato mayonnaise, top bottom bun with onions and mushrooms, close sandwiches, place in press, close gently, and cook for 5 minutes. Press more firmly to compress sandwiches and continue cooking until crisp and fully heated through, about 5 minutes longer. Serve immediately.
