In a large skillet, heat 1 tablespoon of the oil until shimmering. Add the turnips and radishes and cook over moderately high heat, stirring, until lightly browned and crisp-tender, 10 minutes. Add the honey and cook over moderate heat, stirring, until the vegetables are glazed, 5 minutes. Add the soy sauce and cook until syrupy, 5 minutes longer. Add the lemon juice and the Swiss chard; cook until the chard is wilted, 2 minutes. Raise the heat to high and cook until all of the liquid has evaporated, 2 minutes longer; keep warm.
In a medium bowl, whisk the molasses with the water and season with salt. Add the shiitake and toss to coat. Drain the mushrooms, squeezing out most of the excess liquid. In a separate bowl, toss the mushrooms with the rice cracker crumbs, pressing to help the crumbs adhere.
In a large skillet, heat the remaining 1/4 cup of oil until shimmering. Add the coated mushrooms and cook over high heat, turning once, until golden and crisp, 5 minutes. Transfer to paper towels to drain. Top the vegetables with the mushrooms and serve immediately.