Heat a medium sauté pan over medium-low heat, add the olive oil, and heat through. Add the butter and allow to melt, then add the breadcrumbs and stir to evenly coat. Let cook, stirring occasionally, until the breadcrumbs are golden brown, about 2 minutes. Remove to a small bowl and set aside.
For the braised leeks:
Cut the ends and the dark green leaves of the leeks and discard. Slice the leeks into equal 1½-inch (4cm) rounds.
Add to a bowl of cold water and let soak, rinsing any rounds under a running faucet to remove any remaining dirt if needed. Remove and pat completely dry with paper towels.
Heat a large sauté pan over medium heat, add the olive oil, and heat through. Add the shallots and sauté until just tender, about 3 minutes.
Arrange the rounds of leeks into the pan, cut sides down. Cook, shaking the pan until they are golden brown on the bottom, about 3 minutes.
Pour the wine into the pan to deglaze it, lower the heat to medium-low, and cover, cooking until the leeks are tender, about 20 minutes.
Use tongs or a spatula to carefully remove the leeks to a serving dish, reserving as much of the shallots and remaining wine in the pan. Return the pan to medium heat and slowly whisk in the butter, 1 cube at a time. Once it’s all been incorporated, whisk in the cream.
Pour the pan sauce over the braised leeks and finish with the breadcrumbs, and serve.
Notes
Don't skip washing the leeks! Leeks often trap dirt between their layers. Slice them into rounds and rinse thoroughly in cold water before braising.