Combine the vegetable broth and water in a medium saucepan and bring to a simmer over medium heat. Keep the liquid warm on low heat to ensure it doesn't drop the temperature of the barley when added later.

Heat the oil in a large Dutch oven over medium heat until it shimmers. Add the onion and carrot and cook, stirring occasionally, until they are softened and translucent, about 5 to 7 minutes.

Add the barley to the pot and cook, stirring frequently, until the grains are lightly toasted and smell nutty, about 2 minutes.

Pour in the wine and simmer, stirring constantly, until the liquid has been completely absorbed by the grains.

Add about 1 cup of the warm broth mixture to the barley. Simmer over medium-low heat, stirring frequently, until the liquid is absorbed. Repeat this process by adding a ladleful of broth at a time, allowing each addition to be absorbed before adding more.

Continue the addition of broth for 40 to 50 minutes, until the barley is tender but still has a slight chew. You may not need all the broth—stop when the consistency is supple and velvety.

Remove the pot from the heat and stir in the thyme, Parmesan, and butter. Season with salt and pepper to taste, then cover and let sit for 2 minutes to allow the flavors to meld.

Serve immediately while the texture is loose and creamy, offering lemon wedges and extra cheese at the table.