Heat 1 tablespoon oil in 12-inch nonstick skillet over medium-high heat until shimmering. Add poblano and ⅛ teaspoon salt and cook until softened, about 3 minutes. Stir in cumin and cook until fragrant, about 15 seconds. Stir in beans and ½ cup water and cook until beans are heated through, about 4 minutes. Off heat, mash half of beans to chunky paste with potato masher. Stir in lime juice and season with salt and pepper to taste; transfer mixture to bowl and cover to keep warm. Wipe skillet clean with paper towels.

Heat 2 teaspoons oil in now-empty skillet over medium-high heat until shimmering. Add kale and ⅛ teaspoon salt, cover, and cook until kale begins to wilt, about 2 minutes. Stir in ¼ cup water, cover, and cook until kale is tender, 2 to 4 minutes. Using tongs, transfer kale to separate bowl. Wipe skillet clean with paper towels.

Whisk eggs, 2 tablespoons water, pepper, and remaining ¼ teaspoon salt in bowl until well combined. Heat remaining 1 teaspoon oil in now-empty skillet over medium-high heat until shimmering. Add egg mixture and cook, using rubber spatula to constantly and firmly scrape along bottom and sides of skillet, until eggs begin to clump and spatula leaves trail on bottom of skillet, 1½ to 2½ minutes. Off heat, add kale and constantly fold eggs and kale until eggs have finished cooking, 30 to 60 seconds. Cover to keep warm.

Wrap tortillas in damp dish towel and microwave until warm and pliable, about 1 minute. Lay warm tortillas on counter and spread bean mixture evenly across center of each tortilla, close to bottom edge. Top with kale-egg mixture then sprinkle with tomato and drizzle with extra oil to taste. Working with 1 tortilla at a time, fold sides then bottom of tortilla over filling, then continue to roll tightly into burrito. Serve immediately.