Whole-grain crackers and walnuts stand in for breadcrumbs to coat the chicken. Mushrooms add a smoky finish to the stuffing and provide iron, along with the spinach.
Preheat oven to 375 degrees. Pulse crackers and 3/4 cup walnuts in a food processor until finely ground.
Heat oil in a medium skillet over medium heat. Cook garlic and shallots until soft, about 4 minutes. Add mushrooms and thyme, and cook until mushrooms are slightly golden, about 4 minutes; transfer to a plate. Add spinach and water to skillet. Cook until wilted, about 1 minute. Chop spinach, and add to mushroom mixture. Stir in remaining 1/2 cup walnuts.
Season each chicken breast with 1/4 teaspoon salt and some pepper. Lay 1 chicken breast on a plate. Spread 1/2 cup mushroom-walnut mixture onto chicken, leaving a 1/2-inch border all around. Roll, and tie with kitchen twine. Cover with crumb mixture, pressing to coat. Repeat with remaining chicken, mushroom-walnut mixture, and crumbs. Transfer to a rimmed baking sheet, and bake until golden, about 30 minutes. Untie, and slice.