To prepare pesto butter, place first 5 ingredients in a food processor; process until finely chopped. Scrape down sides; add butter. Process until well combined. Cover and chill.
Preheat oven to 375°.
To prepare casserole, combine milk, 1 cup broth, and salt in a large saucepan; bring to a boil. Gradually add grits, stirring constantly. Reduce heat to medium-low; simmer 5 minutes or until thick, stirring occasionally with a wooden spoon. Remove from heat; add 1/2 cup cheese and pepper, stirring until cheese melts. Cool slightly.
Place egg whites in a large bowl; beat with a mixer at high speed until foamy. Add cream of tartar; beat until stiff peaks form. Gently stir one-fourth of egg white mixture into grits mixture; gently fold in remaining egg white mixture. Spoon into a 2-quart soufflé dish coated with cooking spray. Bake at 375° for 40 minutes or until casserole has risen and begins to brown. Serve warm with pesto butter.