In a medium pot, heat 1 tbsp. EVOO over medium-high. Add the onion and chopped garlic and cook 2 minutes, stirring. Add the cucumber and lemon juice and cook until golden. Stir in broth, beans and 1 tbsp. mint. Remove from heat and let cool.
Preheat the broiler. in a blender, puree soup until smooth. With the blender running, slowly pour in 2 tbsp. EVOO. Return to pot. Season and heat over low.
Place the baguette on a foil-lined baking sheet, cut sides up. Brush with EVOO. Broil to toast. Rub the bread with the cut sides of the halved garlic clove, sprinkle with cheese and broil to melt. Top the soup with the tomato and remaining mint. Serve with the cheese toasts.