Heat 1 tablespoon oil in a medium nonstick skillet over medium heat. Add onion and cook, stirring occasionally, until brown, 8 to 10 minutes. Reduce heat to very low, season with 1/4 teaspoon pepper and 1/8 teaspoon salt. Continue cooking, stirring occasionally, until caramelized, about 15 minutes Transfer to a bowl and keep warm.
Meanwhile, place milk in a small bowl and place breadcrumbs on a large plate. Dip each mushroom cap in milk, then dredge in the breadcrumbs.
Heat 1 tablespoon oil in a large nonstick skillet over medium heat. Add mushrooms, gill-side down. Place a heavy, heatproof plate or pie pan on top of the mushrooms and cook until golden brown, pressing down on the plate periodically to flatten them, about 6 minutes. Carefully remove the plate using an oven mitt or tongs, add 1 tablespoon oil to the pan and turn the mushrooms over. Replace the plate and cook, pressing the plate once or twice, until the mushrooms are golden brown and cooked through, 5 to 6 minutes more. Remove from heat; cover to keep warm.
Heat the remaining 1 tablespoon oil in a Dutch oven over medium-high heat. Add garlic and cook until fragrant, 20 to 30 seconds. Add spinach and cook, stirring, until just wilted, about 2 minutes. Stir in beans, broth and the remaining 1/4 teaspoon pepper and 1/8 teaspoon salt. Cook, stirring occasionally, until heated through, 1 to 2 minutes.
Cut each mushroom into thin slices and serve over the spinach. Top with the reserved onions and cheese.