Heat 2 tablespoons of the olive oil in a large saute pan over medium heat. Add the garlic and shallot and cook, stirring frequently, until softened, about 1 minute. Add the mushrooms and wine or stock and simmer until the pan is completely dry, about 7 minutes. Continue cooking until the mushrooms are lightly browned, about 1 minute. Set aside to cool.
Combine the cooled mushroom mixture, ground chicken, breadcrumbs, eggs, basil, salt and some pepper in a large bowl. Blend well and then form into 4 patties. Cook the patties in a saute pan over medium to medium-high heat with the remaining 1 tablespoon olive oil, 4 to 5 minutes per side. (Or cook on the grill, or roast in a 425 degree F oven, until cooked through, about 12 minutes.)
Cut the tomato into 4 thick slices and place on a baking sheet. Top each tomato slice with 1 tablespoon of the grated Parmesan. Split the 4 rolls, spread with softened butter and arrange cut-side up on the baking sheet with the tomatoes. Broil until the buns are toasted and the cheese is slightly melted, keeping a careful watch to prevent burning.
Spread one side of the toasted buns with Dijon mustard before assembling the burgers and topping with lettuce.