Bring a large pot of salted water to a boil. Add the wide egg-free wheat noodles or other dairy-free noodles and cook according to the package directions until just al dente. Drain the noodles, rinse briefly under hot water, and set aside.

While the noodles cook, pat the beef sirloin or stew beef strips dry with paper towels and season lightly with a pinch of the salt and freshly ground black pepper.

Heat 1 tablespoon of the olive oil in a large deep skillet or wide pot over medium-high heat. When the oil is hot, add half of the beef strips in a single layer and sear for 2 to 3 minutes per side until browned but not fully cooked through. Transfer the browned beef to a plate and repeat with the remaining 1 tablespoon olive oil and beef strips. Set the browned beef aside.

Reduce the heat to medium and add the dairy-free butter or margarine to the same skillet. When melted, add the finely chopped yellow onion and cook, stirring often, for 4 to 5 minutes until softened and lightly golden.

Stir in the minced garlic and cook for 30 seconds to 1 minute, just until fragrant, being careful not to let it burn.

Sprinkle the all-purpose flour evenly over the onion and garlic. Stir continuously for 1 to 2 minutes to coat the vegetables and cook off the raw flour taste, forming a thick paste on the bottom of the pan.

Slowly pour in the beef broth while whisking or stirring constantly to dissolve the flour mixture and prevent lumps. Continue to stir until the liquid begins to thicken slightly.

Pour in the full-fat canned coconut milk, then add the soy sauce or tamari, dried thyme, dried oregano, smoked paprika, the remaining salt, and the remaining freshly ground black pepper. Stir well to combine all the seasonings into the sauce.

Return the browned beef and any accumulated juices on the plate to the skillet. Stir to coat the beef in the sauce, then bring the mixture to a gentle simmer over medium heat.

Reduce the heat to low and let the beef and sauce simmer for 15 to 20 minutes, stirring occasionally, until the beef is tender and the sauce has thickened into a silky, gravy-like consistency.

Add the cooked noodles to the skillet with the beef and sauce. Toss gently with tongs or a large spoon until the noodles are evenly coated and everything is well combined. If the mixture seems too thick, you can loosen it with a splash of additional beef broth, added a little at a time.

Taste the beef and noodles and adjust the seasoning with a little more salt or freshly ground black pepper if needed. Sprinkle the chopped fresh parsley over the top, if using, and serve the dairy-free beef and noodles hot.