Cut 2 of the peaches into 1-inch cubes and set aside, then chop the third peach and add to a cocktail shaker. Muddle the chopped peach until it becomes pulpy. Add the muddled pulp to a large pitcher. Reserve the shaker.
Add the Grüner Veltliner, dry vermouth, triple sec, orange juice and lime juice to the pitcher. Add all but a splash of the peach nectar.
In the cocktail shaker, add the jalapeños and the remaining splash of peach nectar. Muddle well to extract the oils from the jalapeño. Empty the contents of the shaker into a coffee filter or cheesecloth and secure with a rubber band to create a sachet (this step allows you to control the level of heat in the cocktail). Steep the sachet in the pitcher for 10 minutes. Remove the sachet and stir. Test the drink for spice and continue to steep the sachet as needed in 10-minute increments until the desired spiciness is reached.
Once the level of heat is to your liking, remove the sachet and use a fine-mesh strainer to strain out the peach pulp. Add the remaining chopped peaches and refrigerate at least 1 hour. The sangria will keep in the fridge for up to 6 days. Serve over ice.