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Line a 15x10x1-inch baking pan with paper towels. Place drained tofu on the paper towels; top with additional paper towels and then top with a baking sheet. Place two or three cans of food on the baking sheet to weigh it down. Chill about 1 hour or until each block of tofu is reduced in thickness by a third. Slice one block of the pressed tofu in half vertically and then horizontally so you have four pieces, each about the size of a deck of cards. Repeat with the remaining tofu block to make eight pieces total.
In a food processor, combine 1 1/2 cups basil, water, 3 tablespoons olive oil, garlic, 2 teaspoons balsamic vinegar, 1 teaspoon kosher salt and 1/2 teaspoon pepper. Cover and process until smooth. Set aside 1/4 cup of the mixture.
Place tofu pieces in a resealable plastic bag set in a shallow dish. Add the remaining basil mixture. Seal bag; turn to coat tofu. Marinate in the refrigerator 1 hour.
Meanwhile, for chutney, in a large skillet heat 2 tablespoons olive oil over medium-low heat. Add onions and fennel; cook about 20 minutes or until golden brown, stirring occasionally. Remove from heat. Stir in 1/2 cup thinly sliced basil, pine nuts, currants, 1 tablespoon balsamic vinegar, 1/2 teaspoon kosher salt and 1/4 teaspoon pepper. Set aside.
Core and slice tomatoes into 1/2- to 3/4-inch-thick slices. You should have about 16 slices. Pat tomato slices dry with paper towels. Season with 1/2 teaspoon kosher salt and 1/4 teaspoon pepper.
Remove tofu pieces from marinade, discarding marinade. Pat tofu dry with paper towels.
Coat a grill pan with cooking spray. Heat pan over high heat. Place tomato slices on grill pan; cook 2 minutes, turning once and changing the direction slices are placed on the grill pan halfway through the cooking of each side (this will produce cross-hatch marks). Transfer tomato slices to a platter. Cook tofu slices on hot grill pan 4 minutes, turning as directed for the tomatoes. Cook tomatoes and tofu in batches if necessary; coat pan with cooking spray as needed.
Serve tomatoes and tofu with the chutney. Drizzle tomatoes with the reserved basil mixture.