Cover eggplant slices on both sides with ground sea salt and place them in a colander, standing up on their sides; let them drain for 30 minutes.
Rinse them under cold water to remove salt and place them on a cutting board lined with paper towels to absorb any excess water. Pat the tops dry – change paper if needed.
Prepare the breading station...In the first shallow dish, add flour, Herbes de Provence and smoked paprika; whisk to blend. In the second shallow dish, add beaten eggs.
In the third dish, add breadcrumbs.
Working with one at a time, coat an eggplant slice in the flour mixture, shaking off any excess flour.
Transfer to the beaten eggs; coat well and hold up to drip off any excess egg.
Coat slice in breadcrumbs and place on a large plate; set aside while doing the others.
Spread butter on bottoms and tops of hamburger buns and place them on a large baking sheet; set aside.
Set the broiler on low.
Transfer the buttered buns under the broiler and grill until golden brown, about 10 to 12 minutes. Remove from the heat and set aside.
In a small skillet over medium heat, add 2 tablespoons of butter. When sizzles, add mushrooms and toss to coat before seasoning with salt and black pepper. Cook for 4 to 5 minutes, stirring often.
Add pressed garlic and sauté for 30 seconds. Transfer the mixture to a bowl and set aside.
In a large skillet over medium-high heat, add canola oil. When it starts shimmering, place coated eggplant slices and fry until browned, about 2:30 minutes. Flip them and do the same with the other side.
Transfer them to a cutting board lined with paper towels to absorb any excess fat.
To assemble the burgers… Spread rémoulade sauce on top and bottom of the grilled buns. Place eggplants on bottom buns, followed by cheese, mushroom mixture, tomato slices, onion rings plus more onion pieces in the middle, and lettuce. Place top buns and serve immediately.