Heat 3 tablespoons oil in a large skillet over medium-high until shimmering. Add half the shallots and cook, stirring, until golden brown and crisp, 3–4 minutes. Using a slotted spoon, transfer shallots to a paper-towel-lined plate; season with salt. Repeat with the remaining shallots; set aside. Leave any remaining oil in the pan.
Add bread to skillet and cook over medium heat until toasted, 3-4 minutes; season with salt and pepper. Transfer to a small bowl.
Add the remaining 1 tablespoon oil to skillet; heat over medium-high. Add mushrooms and cook until starting to brown, about 4 minutes. Season with salt and continue cooking until deep golden brown, about 2 more minutes. Transfer to a small bowl.
Add green beans and ¼ cup water to the pan, cover, and cook over medium heat until tender, 5–6 minutes.
Meanwhile, whisk Dijon, vinegar, garlic, thyme, ½ teaspoon salt, and ¼ teaspoon pepper in a small bowl. Add vinaigrette to beans and cook until fragrant, 30–60 seconds.
Remove from heat and stir in chard, mushrooms, and croutons. Transfer bean and green mixture to a large platter and top with crispy shallots.