Heat 2 tablespoons oil in a medium saucepan over medium heat. Add chopped onion and minced garlic; cook, stirring often, until softened, about 5 minutes. Stir in chopped sun-dried tomatoes; cook, stirring constantly, until fragrant, about 1 minute.
Add chopped artichoke hearts, cannellini beans with liquid, 8 cups broth, ¾ teaspoon salt and ½ teaspoon pepper; bring to a boil over medium-high heat. Reduce heat to maintain a lively simmer; cook, stirring occasionally, until the flavors start to meld, about 5 minutes.
Add 6 cups kale and 1 cup peas; cook, stirring occasionally, until bright green and tender, about 2 minutes. Stir in 1½ cups chicken and 1½ tablespoons lemon juice. Sprinkle with ¼ cup Asiago. Garnish with additional pepper, if desired.
