Step 1

Remove and reserve a few darker outer leaves from the lettuce, discarding any blemished ones. Halve the hearts lengthwise. Rinse, drain and thinly slice the reserved outer leaves.

Step 2

In a wide, heavy pot over medium-high heat, heat 1 tablespoon of the oil until shimmering. Add the garlic and cook, occasionally flipping and stirring the cloves, until golden brown, 1 to 2 minutes. Use a slotted spoon to transfer the garlic to a small bowl.

Step 3

Add the lettuce heart halves to the pot, cut side down (don't be alarmed if they spit and pop a bit), and cook until golden brown, 1 to 2 minutes. Turn them over, add 1/4 teaspoon of the salt and cook just until they've gone translucent at the stem and wilted at the tips, about 3 minutes more. It's okay if a few leaves fall off. Transfer the halved hearts to a plate. Add the sliced lettuce and another 1/4 teaspoon of salt to the pot, and cook just until wilted and almost all the liquid in the pot has evaporated, about 2 minutes. Transfer to a medium bowl.

Step 4

Reduce the heat to medium-low and add the remaining 1 tablespoon of oil and the butter. When the butter froths, add the onions, the cooked garlic and 1/4 teaspoon of the salt, then stir and cover the pot. Cook, stirring occasionally, until the onions are soft and creamy but not colored, about 20 minutes. Increase the heat to high, pour in the wine and boil until it's nearly all evaporated, about 5 minutes.

Step 5

Add the peas and return the cooked sliced lettuce to the pot; stir well, then return the lettuce heart halves, cut side up, to the pot along with the broth and the remaining 1/4 teaspoon of salt. Bring to a boil, then reduce the heat to maintain a steady simmer, prodding the lettuce and peas a bit so they're nearly submerged. Cook just until the peas are soft but still bright green and the flavors marry, 5 minutes.

Step 6

Add the mint and a healthy glug of oil, and stir. Season lightly with salt.

Step 7

Serve warm, spooned over the cooked barley.