Cut tofu into 10 thin slices. Set on a baking sheet, cover with another baking sheet, and weigh down with something heavy, such as a large can of food. Drain until excess water is removed, at least 30 minutes.
Heat olive oil in a small cast iron skillet over medium heat. Toast ground cumin, chili powder, and oregano in the hot oil until spices are aromatic, about 2 to 3 minutes.
Transfer oil and spice mixture to a saucepan over medium heat; stir in minced garlic, white wine vinegar, and white wine. Bring to a simmer, then remove marinade from heat.
Place drained tofu into a glass dish; pour marinade over tofu. Cover with plastic wrap; refrigerate for at least 8 hours.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Remove tofu from marinade; pat dry with paper towels and place on broiler rack. Broil on each side until skin is crisp, about 6 minutes per side.
To assemble, spread mayonnaise on one side of each slice of bread; place 2 tofu slices on 5 of the bread slices; top tofu with sliced tomato, arugula, red onion, salt, black pepper, and remaining slices of bread.