Combine milk, flour, and eggs in a blender. Blend until smooth, about 1 minute.

Heat a nonstick pan over medium-high heat. Brush hot pan with a small amount of butter. Ladle about 3 tablespoons of batter into the center of the pan. Tilt pan from left to right to cover entire surface. Sprinkle top of the crepe with 1 or 2 teaspoons of basil.

Cook until edges begin to brown, 2 to 3 minutes. Flip crepe over and cook until set, about 2 minutes more. Remove to a plate. Cook remaining batter, re-brushing pan with butter and stacking crepes with parchment paper in between.

Mix chicken, grapeseed oil, balsamic vinegar, and salt together in a resealable zip-top bag; marinate for 20 to 30 minutes.

Drain and discard marinade. Heat a skillet over medium-high heat. Add chicken; cook and stir until juices run clear, about 5 minutes.

Place broccoli in a microwave-safe bowl with 2 tablespoons water. Microwave until crisp-tender, 3 to 4 minutes.

Combine butter and garlic in the same skillet over medium heat. Cook until fragrant, about 1 minute. Add the broccoli; stir to coat with the garlic butter. Stir in tomatoes and mushrooms and heat through.

Stir cream, cheese, and pesto together in a separate saucepan over medium heat. Cook and stir until sauce comes together, 3 to 5 minutes.

Lay a crepe on a plate, line chicken filling down the middle, and drizzle sauce on top. Fold crepe over burrito-style; spoon more chicken and sauce over top. Repeat with remaining crepes, filling, and sauce.