Heat the oil in a small saucepan over medium-high heat. Add the garlic, onion and carrot and cook, stirring often, until light golden brown at the edges, about 3 minutes.
Reduce the heat to medium and add the jalapeños, bay leaf, oregano, piloncillo and salt. Cook, stirring until the sugar melts, about 1 minute. Add the vinegar and ¼ cup water and bring to a boil, stirring. Reduce the heat to low and simmer, stirring occasionally, until the jalapeños are khaki green and almost tender, 2 to 3 minutes. Remove from the heat.
Transfer to a bowl to serve immediately or to a jar to use later. To jar, carefully spoon the solids into a pint jar, then pour in the liquid. If needed, add enough water to barely cover the solids. Screw on the lid fingertip-tight, then cool to room temperature. Refrigerate until ready to use.
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