[For best flavor, sauté the vegetables before adding to the slow cooker. If you’re short on time, this step can be skipped.]
Heat the oil in a medium sauté pan or skillet over medium-high heat. When hot, add the shallots, carrots, celery, and a pinch of salt, and cook until the vegetables are beginning to soften, about 5 minutes. Add the garlic and mushrooms, and cook for an additional 4 to 5 minutes, until the garlic is fragrant and the mushrooms have released their liquid.
Transfer the cooked vegetables to a large (5 to 6-quart capacity) slow cooker, then add the potatoes, beans, canned tomatoes, broth, herbs, salt, and a few grinds of black pepper. Stir, cover, and set to cook on high for 4 to 6 hours, or low for 8 to 10 hours. Taste and adjust seasonings as necessary. Just before serving, stir in the fresh parsley.
The cassoulet will keep in the refrigerator for up to five days. Cooled portions can be frozen for up to three months.