Heat 2 tablespoons oil in a large skillet over medium-high heat. Add mushrooms and onion; cook, stirring occasionally, until starting to brown, 5 to 7 minutes. Add garlic and cook, stirring, for 30 seconds. Add collards and water; reduce heat to medium and cook, stirring occasionally, until the collards are tender, 5 to 7 minutes. Season with pepper and salt.

Preheat a panini maker.

Brush one side of each slice of bread with some of the remaining 2 tablespoons oil. Turn 4 pieces over and spread 1 tablespoon Dijonnaise on each. Top with one-fourth of the collards mixture (about 3/4 cup), a slice of cheese and the remaining bread, oiled-side up.

Cook the sandwiches in the panini maker until golden brown and the cheese is melted.