Heat the oil, garlic, and rosemary sprigs in a small saucepan over medium heat until they start to sizzle steadily and become fragrant, 2 to 3 minutes. Let the oil cool to room temperature. Strain the oil into a measuring cup. (If making ahead, store in the refrigerator and use within 3 days.)
Put the tomatoes in a medium bowl and toss with 3 Tbs. of the garlic oil, the mint, and 1 tsp. of the salt.
Put the beans in a food processor and add about 6 Tbs. of the garlic oil, the Parmigiano, 1 Tbs. of the lemon juice, the chopped rosemary, remaining 1/2 tsp. salt, and the black pepper and puree until smooth. Season to taste with more salt, pepper, and lemon juice.
Heat a gas grill to medium high or prepare a medium-hot charcoal fire. Brush both sides of the bread with the remaining garlic oil. (If you run out, use plain olive oil to finish.) Sprinkle lightly with salt. Grill the bread until crisp, with nice grill marks on both sides, 1 to 2 minutes per side.
Spread the grilled bread with the bean purée, top with a generous spoonful of the tomatoes and their juices, sprinkle lightly with pepper, and set out on a large platter so your guests can help themselves.