Step 1Heat the oven to 375 degrees. Lightly brush a shallow 12-by-7-inch baking dish with olive oil.
Step 2Halve each pepper lengthwise, taking care to leave the base and stem intact (the peppers should look like little boats) for each of the larger halves. Chop the remaining smaller halves into a small dice and set aside.
Step 3Remove the seeds, then cook the peppers in salted, boiling water just until tender, 6 to 8 minutes. Strain and set aside.
Step 4In a large, heavy skillet heat 1 tablespoon oil. Add the onion and sauté until just tender, 2 to 3 minutes. Add the mushrooms and sauté until they are tender and have released their juices, 3 to 5 minutes, stirring occasionally.
Step 5Add the diced pepper and squash and sauté until just tender, 3 to 5 minutes more, adding oil as needed to prevent sticking. Add the corn kernels and 1 teaspoon each of minced basil and chopped dill. Add one-half teaspoon salt and one-eighth teaspoon pepper. Remove from the heat and set aside.
Step 6Make the tomato sauce, by coarsely chopping the tomatoes and pulsing them in a food processor to a coarse purée. Heat 1 tablespoon olive oil in a small sauté pan over medium heat and add 2 cloves minced garlic, sautéing just until aromatic. Add the tomato purée and half of the chicken broth.
Step 7Add one-fourth teaspoon salt and bring just to a boil, then reduce the heat and simmer for 15 minutes. Halfway through, stir in the remaining chicken broth. Remove from heat and stir in the roughly chopped basil.
Step 8To assemble, spoon the sauce into the prepared baking dish, covering the bottom completely and evenly. Divide the sautéed vegetables evenly among the pepper halves, filling the peppers completely.
Step 9In a small bowl, mix the bread crumbs with the remaining garlic, minced basil and chopped dill. Season with a pinch of salt, and stir in 1 tablespoon of the melted butter. Spoon the bread crumbs evenly over the stuffed peppers and drizzle the remaining butter over each.
Step 10Carefully place the peppers in the pan over the tomato sauce. Place the pan in the oven, and bake until the bread crumbs are browned on top and the sauce is bubbly, 25 to 30 minutes. Serve immediately.