In a bowl, whisk together egg yolks, lemon juice, garlic, anchovy paste, mustard, olive oil, 2 tbsp of the vegetable oil and 1/4 tsp of the salt until well combined. Stir in capers and set aside.
In a sauté pan, heat remaining 1 tbsp vegetable oil over medium heat. Add bread crumbs and stir for 2 minutes, until golden brown. Cool.
Toss kale with dressing, remaining 1/8 tsp salt and the pepper. Gently fold in bread crumbs and Parmesan.