Heat two tablespoons oil in a large skillet. Cook leeks for five minutes, turning to prevent burning. Add stock and cover, braising leeks on medium-low heat for ten more minutes.
Drain leeks. Brush with remaining two tablespoons of oil; sprinkle with salt and pepper. Heat grill and place leeks on it, cut side down. Cook until surface starts to char, approximately five minutes. Carefully toss all in a bowl with lemon juice.
To serve, arrange equal portions of leeks. Top with one teaspoon peas, tendrils, and caviar each.