Crack the eggs into a bowl and whisk lightly with a pinch of salt. Thinly slice the cabbage, shred the carrots if needed, mince the garlic, grate the ginger, and slice the scallions. Combine the soy sauce, hoisin, sesame oil, rice vinegar, and sriracha in a small bowl and set aside.
Heat a large, heavy skillet over medium-high heat and add the avocado oil. When the oil shimmers, add the ground turkey and cook, breaking it up with a spatula, until it's browned with no pink remaining, about 6 to 8 minutes. Season lightly with half the prepared sauce while cooking to begin flavoring the meat. Transfer the turkey to a bowl and wipe the skillet clean if there's excess fond.
Return the skillet to medium heat and pour in the beaten eggs. Let them sit undisturbed for 20 seconds, then gently push the set edges toward the center to create soft curds. Cook until just barely set, about 1 minute, then transfer the scrambled eggs to the bowl with the turkey.
Add the toasted sesame oil to the empty skillet over medium heat, then add the garlic and ginger. Sauté until fragrant, about 30 seconds. Add the shredded carrots and cook 1 minute, then add the cabbage and cook, stirring occasionally, until the cabbage has softened but still holds a little bite, about 4 minutes. Press the vegetables down firmly with a spatula a couple of times to get a little caramelization.
Return the browned turkey and scrambled eggs to the skillet with the vegetables. Pour the remaining sauce over everything and toss to combine. Cook another 2 to 3 minutes, stirring, until the sauce is mostly absorbed and the flavors have melded. Taste and adjust with a splash more soy or a squeeze of lime if needed.
Remove the skillet from the heat and stir in the sliced scallions, reserving a few green tops for garnish. Let the bowl rest in the pan for 2 minutes so the flavors settle and any excess moisture evaporates.
Divide the egg roll mixture among 4 bowls. Sprinkle with toasted sesame seeds and the reserved scallion slices. Serve with lime wedges on the side for a bright finish.
Leftovers: cool completely, then store in an airtight container in the refrigerator for up to 4 days. Reheat gently in a skillet over medium heat to avoid drying out the eggs.