Prepare the tomato base: If starting from fresh tomatoes, chop about 9 pounds of plum tomatoes and add them directly to a pot over medium-high heat. Cooking immediately after chopping helps prevent separation. Continue chopping and adding tomatoes to the pot until all are added. Simmer for 5 minutes, then pass through a food mill to remove skins and seeds. You should end up with approximately 13 cups of fresh tomato puree (passata).
Alternatively, you can use about 7½ cups of tomato sauce if starting with canned tomato products.
Prepare jars and canner: Before you begin cooking down the sauce, prepare a boiling water bath canner. Wash jars, lids, and bands. Keep jars hot until ready to fill.
Cook the tomato sauce: Pour half the tomato puree into a large stainless steel pot. Bring to a boil over high heat, stirring regularly. Once boiling, add the remaining puree 1 cup at a time, allowing it to return to a boil between additions. Reduce to a simmer and continue cooking until the mixture is reduced by about half. This should take approximately 30 minutes, depending on your stove and pot.
If using canned tomato sauce, cook for about 8 to 10 minutes to slightly reduce volume.
Add the seasonings: Stir in the lemon zest and lemon juice, garlic, sugar, vinegar, salt, Worcestershire sauce, dry mustard, cayenne, onion powder, and black pepper. Do not add the horseradish yet! Increase heat and return the mixture to a full rolling boil, stirring frequently to prevent scorching.
Add horseradish (off heat): Once boiling, immediately turn off the heat and stir in the finely grated horseradish. Do not return to heat after this step—adding the horseradish off-heat helps preserve its sharp flavor and heat.
Fill jars: Ladle hot cocktail sauce into hot jars, leaving ½ inch headspace. Remove air bubbles, wipe rims, apply lids, and screw on bands until fingertip-tight.
Process jars: Place jars in the boiling water bath canner. Make sure water covers the tops of jars by at least 1 inch. Process for 15 minutes (adjust for altitude below). When time is up, turn off heat and let jars rest for 5 minutes before removing to a towel-lined counter. Let cool undisturbed for 12–24 hours. Check seals, label, and store.