Combine jalapeños, spices, vinegar, garlic, and carrot slices in a 2-quart saucepan over medium heat.
Simmer until the peppers are soft, about 15 minutes. Remove peppers from brine and set aside to cool slightly. Keep brine warm on the stove while you stuff the peppers.
Once peppers are cool enough to handle, make a slit vertically down one side of each pepper and carefully remove the seeds and core with a small spoon being careful not to tear the pepper.
Divide the cheese evenly between the pepper and press to close. Gently place peppers in a clean, sterilized jar with a tight-fitting lid, layering them with the carrots and garlic, being careful not to press down on the peppers.
Pour warm brine over the peppers and fill to within 1/4-inch from the top. Seal jar and refrigerate. Let sit at least 1 day before opening. Peppers will keep refrigerated for 1 month.