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Preheat oven to 375°F. Heat 3 tablespoons oil in a large, deep skillet over medium-high. Add mushrooms; cook until browned on bottom, 5 to 7 minutes. Stir; cook, stirring occasionally, until browned all over, 4 to 5 minutes.
Reduce heat to medium. Add remaining 1 tablespoon oil, shallot, thyme, pepper, and ¾ teaspoon salt to skillet; cook, stirring, until softened, 1 to 2 minutes. Stir in kale; cook, stirring often, until wilted, about 2 minutes. Add flour and stir until evenly coated. Add milk, stirring and scraping bottom of skillet, until combined. Add beans, mustard, and ¼ teaspoon salt; bring to a simmer.
Remove skillet from heat. Form 6 wells in mixture. Crack 1 egg into each well; sprinkle with remaining ¼ teaspoon salt. Transfer skillet to oven and bake until egg whites are fully set and yolks are set to desired doneness, 10 to 15 minutes. Sprinkle with cheese and pepper.