Adjust oven rack to upper level and preheat broiler on low.
Cook farro in vegetable broth according to package directions.
De-stem the kale and roughly chop.
Heat 1/4 cup olive oil in a large ovenproof skillet over medium-high heat. Add mushrooms and salt and cook, stirring occasionally, until browned.
Add the carrot and onion and cook, stirring often, until tender - about 5 minutes. Add the kale and cool until slightly wilted, about 3-4 minutes.
Add the wine and stir to loosen any browned bits on bottom of pan. Simmer until liquid has evaporated, about 1 minute. Sprinkle the flour over the mixture, stirring constantly.
Gradually add 1 1/4 cup vegetable broth, 1/2 cup at a time, stirring to incorporate. Bring mixture to a simmer, and cook, stirring often, until thickened, about 2 minutes. Removc from the heat and stir in the parmesan and cooked farro.
Toss the breadcrumbs with 2 tablespoons of olive oil and sprinkle over the gratin.
Broil for 10 minutes, checking after the first 5, until breadcrumbs are golden brown and casserole is bubbling.
Remove from oven and let stand for 10 minutes. Sprinkle with parsley before serving.
*To prepare ahead, skip broiling and cover with foil; refrigerate until ready to bake. Bake at 400°F for 30 minutes until breadcrumbs are golden brown and filling is bubbling.