Heat the olive oil in a large skillet over medium-high heat. Add the onion and bell pepper and cook, stirring, until softened, 3 to 5 minutes. Stir in the garlic, ginger and tomato paste. Cook, stirring, for 1 minute.
Add the beef and season with salt and black pepper. Cook, breaking up the meat with a wooden spoon, until browned, about 3 minutes.
Give the sauce a whisk, then pour it into the meat mixture. Use a wooden spoon to scrape up any brown bits from the bottom of the skillet. Bring the sauce to a boil, then reduce the heat to a simmer. Cook until slightly thickened, about 3 to 5 minutes. Add a dash or two (or three or four...) of sriracha, to taste.
Remove the skillet from the heat and stir in the lime zest, lime juice, and cilantro. Taste and season with additional sriracha and/or lime juice, if desired.