In a small bowl, combine sugar, garlic, salt, and black pepper. Rub the meat on all sides with this mixture. Wrap the fillets in plastic wrap and refrigerate for two hours.
Turn on the oven to preheat to 220 degrees. Clean the head of garlic only from the top layer of the husk, without disassembling it into cloves. Cut off the top of the head and brush with vegetable oil. Wrap the garlic in foil and put in a preheated oven, bake the garlic for about 30-35 minutes, until soft. Chill the baked garlic at room temperature and then grind in a kitchen processor with yogurt and mayonnaise until smooth. Transfer the mixture to a bowl, cover, and refrigerate.
In a large skillet over medium-high heat, heat the olive oil, add the onions and fry, stirring occasionally, for about 5 minutes, then reduce the heat to low and cook the onions for another 10 minutes, until soft. Add mushrooms to the pan with onions and cook until mushrooms are soft. Then add vinegar, sugar, salt, black pepper to the pan, cook for about 1 minute.
Grease the grill rack with oil and turn on the grill to preheat to medium temperature. Place the fillets on an oiled grill rack and fry for about 5-6 minutes on each side (the meat thermo should show a temperature of 160-170 degrees). Let the fried meat rest on the board for about 10 minutes, and then cut into 4 pieces.
Grease the baguette slices with garlic sauce, put green leaves on the sauce, meat on top of the greens, and onion-mushroom mixture on top. Serve the sandwiches right away.
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