With a paring knife, cut two slits on opposite sides of each jalapeño, starting about ½ inch from the stem end and ending about 1 inch from the tapered end.

Heat the oil in a deep non-aluminum saucepan over medium-low heat. Add the jalapeños, onions, carrots, and garlic and cook, stirring often, until the vegetables are softened but not browned, about 10 minutes.

Add the vinegar, water, salt, bay leaves, thyme, marjoram, lime, sugar, mustard, allspice, and peppercorns. Bring to a boil, then adjust the heat so the liquid is simmering. Taste and add a little more water if the mixture seems to sharp to you. Remember, though, that it will mellow somewhat as it sits. Simmer for 15 minutes. Remove the pan from the heat and cool to room temperature.

Pack the vegetable and their liquid into clean glass jars and refrigerate. They will keep for up to 6 weeks, as long as they are kept refrigerated and all the vegetables are covered in liquid.