Rinse your tomatoes under cool running water and pat them dry with a clean towel. Remove the stem end from each tomato by cutting a small circle around the core, then discard any bruised or damaged sections.

Cut each tomato into quarters, working over a large bowl to catch any juices that run out. Aim for roughly equal-sized pieces so they cook evenly.

Transfer the chopped tomatoes and any collected juices into your large pot. Add them gradually so the pot doesn't overflow, stirring occasionally as the tomatoes begin to release more liquid.

Turn the heat to medium-high and bring the tomatoes to a gentle simmer, stirring occasionally to prevent sticking on the bottom. This process takes about 10 minutes and allows the tomatoes to soften and break down.

Simmer the tomatoes for another 15 to 20 minutes, stirring every few minutes, until they're completely broken down and appear almost mushy. The mixture will become darker and more concentrated as the water content reduces slightly.

Working in batches, pour the cooked tomato mixture through a food mill fitted with the finest disk, catching the juice in a large bowl below. If you don't have a food mill, pour the mixture through a fine mesh strainer lined with cheesecloth, pressing gently on the solids with the back of a wooden spoon. Save the juice and discard the seeds and skins left behind.

Pour the strained juice into a clean pot and add the kosher salt, black pepper, and lemon juice. Stir well and taste the juice, adjusting the seasonings as needed by adding more salt for depth, more lemon for brightness, or a tiny pinch of cayenne for subtle heat.

For silky-smooth juice, pour the seasoned liquid through a fine mesh strainer one more time, which removes any remaining fine particles. This step is optional if you prefer a thicker consistency with more body.

Let the juice cool to room temperature, then transfer it to clean glass jars with tight-fitting lids. Store in the refrigerator immediately if drinking within one week, or freeze for longer storage.