Make the pesto. Combine basil, mint, thyme leaves, walnuts, garlic, and 2 tbsp of the olive oil in a food processor until a paste forms. With the processor running, gradually add the remaining 2 tablespoons of olive oil and process until the paste is thinner, smoother, and paler. Add the Parmesan cheese and pulse until just incorporated. Taste and adjust the season, as needed.
In a large skillet, heat 3 tablespoons of the olive oil over medium-low heat. Add bread cubes and 1 teaspoon salt and stir until the cubes are evenly coated. Cook, stirring occasionally, until golden brown, about 10 minutes. Transfer to a serving bowl and wipe out the pan.
In the same skillet, heat the remaining tablespoon of olive oil over medium-low heat. Add the leek and pinch of salt and cook, stirring frequently, until the leek starts to break down, 5 to 7 minutes. Be careful not to cook the leek too quickly as you want it to disintegrate slowly.
Add the asparagus and sauté, stirring frequently, until bright green, 3 to 4 minutes. Add the peas and snow peas and cook, still stirring, until bright green, 2 to 3 minutes more. They should be cooked, but crisp.
Pour the vegetables onto the bread. Add half of the pesto, the vinegar, lemon juice, and Parmesan cheese. Season with pepper and toss together until all vegetables are coated with the pesto. Add more pesto if desired. Garnish with shaved Parmesan, if desired.