Bring the stock to a low simmer in a saucepan. Heat the olive oil in a large saucepan over medium-high heat until hot. Add the garlic and cook briefly until light brown. Add the onion and a pinch of salt, lower the heat to medium, and cook until the onion is soft but not brown, about 2 to 3 minutes. Add 2 cups of the stock and bring to a boil. Add the pastina, stir well, and reduce the heat to maintain a slow simmer. Season with salt and pepper.

Simmer, ladling in the stock 1/2 cup at a time as the previous addition is absorbed and stirring occasionally to prevent the pastina from sticking to the bottom of the pan, until the pasta is about three-fourths done, about 10 minutes. Add the peas and the remaining stock and cook until the peas are tender and the pasta is al dente, about 3 minutes. Remove from the heat and add 1 tablespoon of the butter, the oregano, thyme, Parmesan, and 3 tablespoons of the bacon. Stir well. The consistency should be quite loose, like a thick soup.

During the last few minutes of cooking, warm the carrot juice and spice rub in a small pan over medium heat. Do not allow it to get too hot or solids will form. Whisk in the remaining 1 tablespoon butter until just melted.

Plates are better than soup bowls or rimmed soup plates here, because they allow the carrot juice to spread out around the pastina.

To serve, pour about 2 tablespoons carrot juice onto each of 4 warmed dinner plates. Spoon the pastina into the center of the plates, garnish with a light sprinkling of the remaining bacon, and serve immediately.