Rinse split peas and mixed grains a few times. Set aside.
Heat 2 tablespoons butter in a 3-quart (2.8 liter) dutch oven over medium heat until melted. Add garlic and onion. Stir and cook until the onion turns transparent and golden, about 5 minutes.
Add split peas, mixed grains, chicken stock, and 4 cups water. Mix well.
Bring to a boil over high heat. Carefully skim the foam from the surface and discard it, until the broth is clear. Cover and cook over medium low heat until the peas turn tender, about 50 minutes, stirring occasionally.
While cooking the soup, heat the remaining tablespoon of butter in a skillet until melted. Add bread cubes. Stir and cook until the bread is crispy and evenly browned. Transfer to a small bowl.
Heat olive oil in a skillet until warm. Add rosemary and thyme. Gently cook them until just sizzling. Transfer to a small bowl.
When the peas become tender, blend the soup with an immersion blender until it develops a consistent and creamy texture.
Continue boiling the soup at medium heat, uncovered, until it reaches the desired thickness, 10 to 15 minutes. Add oyster sauce and mix well.
Transfer soup to individual serving bowls, top with crouton and swirl with aromatic oil.