Toast flour in a large skillet over medium-high heat, stirring frequently, until golden brown and smells like toasted nuts, about 8 minutes. Transfer to a plate and wipe out skillet.
Cook onion, poblano chile, in same skillet, stirring and scraping up any browned bits, until softened, about 10 minutes. Add garlic and cook until fragrant. Add tomatoes, season with salt, and continue cooking until most of the liquid has evaporated and tomatoes begin to stick to the skillet, about 8 minutes. Add red chile sauce and cook, stirring frequently, about 1 minute. Add reserved beef and toasted flour and toss until coated.
Whisking constantly, add broth, bring to a boil, and continue to whisk until no lumps remain. Reduce heat to medium low and cook until chili gravy has thickened and beef is tender, 30 minutes; season with salt. Strain through a fine-mesh sieve into a large bowl; set beef and gravy aside separately.
Meanwhile, preheat oven to 400°. Heat remaining 1 cup oil in a medium skillet over medium-high until oil bubbles immediately when edge of tortilla touches the surface. Working one at a time, fry tortilla, turning once, until just starting to brown and crisp, about 10 seconds per side. Transfer tortilla to paper towels to drain. Repeat with remaining tortillas.
Spread 1 cup gravy lengthwise down the center of a 13x9" baking dish. Dip both sides of each tortilla in chili gravy just to coat, then transfer to a plate. Working one at a time, spread ¼ cup beef mixture down the center of tortilla and fold one side over filling, then continue to roll enchilada onto itself. Place seam side down in prepared baking dish as you go (enchiladas should be nestled right up against each other). Top with cheese and remaining gravy. Bake until sauce is bubbling and cheese is beginning to brown, 20–25 minutes. Let sit 10 minutes before serving.
Garnish with pico de gallo and sliced radish.