Preheat oven to 400°F. Toss together fresh breadcrumbs, 1/4 cup of the fresh sage leaves, 2 tablespoons of the oil, and 1/4 teaspoon of the salt on a rimmed baking sheet. Bake in preheated oven until breadcrumbs are golden and sage leaves are crisp, about 5 minutes. Set aside.
Increase oven temperature to broil with oven rack 6 inches from heat. Toss together tomatoes and 1 tablespoon of the oil on a rimmed baking sheet. Broil until tomatoes are blistered and beginning to collapse, 5 to 7 minutes, shaking sheet halfway through broiling.
Meanwhile, bring a large pot of salted water to a boil. Cook pasta until al dente, about 8 minutes. Drain, reserving 1 cup cooking water.
Heat 1 tablespoon of the oil in a very large skillet over medium. Add onion and garlic; cook, stirring occasionally, until soft and golden, about 10 minutes. Add beans, broth, and remaining 1/4 cup each sage leaves and oil; bring to a simmer. Cook, stirring occasionally, 10 minutes. Gently stir in blistered tomatoes. Simmer, stirring occasionally, 10 minutes.
Add cooked pasta and reserved cooking water to bean mixture; toss gently to combine. Stir in pepper and remaining 1/4 teaspoon salt. Cook, stirring often, until liquid thickens slightly, about 4 minutes. Top with crispy breadcrumb mixture. Drizzle with oil, and serve immediately.