Step 1

Toss shallot, thyme, mushrooms, oil, vinegar, and salt in a medium bowl. Let sit at room temperature 1 hour.

Step 2

Heat oil in a large skillet over medium heat. Cook onion and garlic, stirring often, until onion is softened and just translucent and garlic is golden, about 4 minutes.

Step 3

Working in batches, add kale, tossing and letting it wilt slightly before adding more. Reduce heat to medium-low and cook kale, tossing often, until tender, 12–15 minutes. Mix in red pepper flakes and season with salt and pepper.

Step 4

Place dried mushrooms in a small bowl and cover with boiling water; let stand until rehydrated and pliable, 5–8 minutes. Drain, squeeze out excess moisture, and finely chop.

Step 5

Mix mushrooms, shallot, mayonnaise, lemon juice, and rosemary in a small bowl; season with salt and pepper. Cover and chill.

Step 6

DO AHEAD: Mayonnaise can be made 3 days ahead. Keep chilled.

Step 7

Heat oil in a medium skillet over high heat. Drain marinated mushrooms, pat dry, and cook until deep golden brown, 6–8 minutes per side.

Step 8

Meanwhile, heat butter in a medium nonstick skillet over low heat and cook eggs, stirring occasionally with a heatproof spatula, until just barely set, about 4 minutes (or cook over easy, if you prefer).

Step 9

Heat broiler. Spread cut sides of rolls with mushroom mayo (you may have some left over). Place rolls, cut side up, on a broilerproof rimmed baking sheet. Layer bottom halves of rolls with marinated mushrooms, kale mixture, scrambled eggs, and cheese. Broil until cheese is melted.

Step 10

Close sandwiches and serve with hot sauce, if desired.