Put the chicken in a large bowl and set aside. Heat 2 T oil in a large skillet over medium heat. Add the onions and a pinch of salt and sauté until translucent. Add the garlic and sauté for a couple more minutes, just until fragrant. Scrape the onions and garlic into the bowl with the chicken. Add the mushrooms to the pan, season with salt and pepper, and sauté until softened, about 8 to 10 minutes. Add to the bowl with the chicken and onions.
Meanwhile, combine milk, chicken broth, minced thyme and dried mushrooms in a medium saucepan. Bring almost to a boil and then turn off the heat and let stand about 5 minutes to allow the flavors to come together. If the dried mushrooms are large, left them from the milk mixture with a slotted spoon and mince, then return them to the mix.
Melt the butter in a Dutch oven over medium-high heat. Whisk in the flour and some salt and pepper and continue to whisk until the flour is starting to cook, 3 or 4 minutes. Add the hot milk mixture all at once, continuing to whisk until smooth. Bring to a simmer, whisking until the mixture thickens, like a cream soup. Stir in the cheese.
Measure out about 1½ cups of the sauce and stir it into the chicken mixture. Taste and adjust seasoning to your liking. The chicken and the extra sauce can both be refrigerated, covered, at this point for 2 days.
Heat oven to 325. Line a rimmed baking sheet with parchment paper. Take 1 crepe at a time and spread about ¼ cup of filling across the bottom third of the crepe. Roll into a fairly tight cylinder, and set on baking sheet. When they’re all assembled, bake crepes, uncovered, until warm and golden, about 15 to 20 minutes. Warm the remaining sauce (it should be quite hot) thinning with a little milk or water. Stir in parsley. Spoon the sauce over the crepes and serve.