1. Adjust oven rack to upper-middle position and heat oven to 450 degrees. Heat oil in 12-inch nonstick skillet over medium heat until shimmering. Add onion and cook until softened, about 5 minutes. Stir in beef, breaking up meat into small pieces, and cook until no longer pink, 8 to 10 minutes. Using slotted spoon, transfer beef mixture to paper towel–lined plate. Pour off all but 2 tablespoons fat from skillet.
2. Add poblanos and cook over medium-high heat until browned, 8 to 10 minutes. Stir in beef mixture, garlic, cumin, ¾ teaspoon salt, oregano, ½ teaspoon pepper, and cayenne and cook until fragrant, about 30 seconds. Add tomatoes and cook until beef mixture is dry, about 1 minute. Off heat, stir in 2 cups Monterey Jack. Scrape mixture into 13 by 9-inch baking dish, pressing into even layer.
3. Combine flour, remaining ½ teaspoon salt, and remaining ¼ teaspoon pepper in bowl. Slowly whisk milk into flour mixture until smooth; set aside. Using stand mixer fitted with whisk, whip egg whites on medium-low speed until foamy, about 1 minute. Increase speed to medium-high and whip until stiff peaks form, about 3 minutes. Whisk one-third whipped egg whites into batter, then gently fold in remaining whites, 1 scoop at a time, until combined.
4. Pour batter over beef mixture. Bake until topping is light golden and puffed, about 15 minutes. Sprinkle with remaining ½ cup cheese and bake until golden brown, about 10 minutes. Let cool on wire rack for 10 minutes. Serve.