Step 1

Whisk garlic, vinegar, miso, mustard, and 3 Tbsp. water in a small bowl to combine; set sauce aside.

Step 2

Combine walnuts and 2 Tbsp. oil in a large skillet, preferably cast iron, and set over medium heat. Cook, stirring often, until nuts are golden brown, about 5 minutes. Using a slotted spoon, transfer to a large plate and season with salt.

Step 3

Wipe out skillet and add 1 Tbsp. oil, then sausage. Cook over medium-high heat, undisturbed, 3 minutes. Stir and continue to cook until sausage is mostly crisp and rust-colored, about 3 minutes longer. Using slotted spoon, transfer to plate with walnuts.

Step 4

Add half of cabbage to skillet and stir once to coat in oil. Cook (still over medium-high), undisturbed, until charred underneath, about 4 minutes. Add remaining cabbage, stir, and cook, stirring occasionally, until wilted, 6–8 minutes. Season with salt and lots of pepper. Return walnuts and sausage to pan; add sliced celery and reserved sauce. Cook, stirring, until coated, about 30 seconds. Remove from heat; season with more salt and pepper if needed.

Step 5

If topping with an egg, heat remaining 2 Tbsp. oil in a large nonstick skillet over medium-high. Crack eggs into skillet and season with salt and pepper. Cook until edges are browned and crisp and whites are just set but yolks are still runny, about 5 minutes.

Step 6

To serve, stir dill and celery leaves into stir-fry. Divide among plates and top each with a fried egg if desired.