In a medium saucepan, heat 3 inches of grapeseed oil over medium-high heat until the oil reaches 350° on a digital thermometer. Add the hazelnuts and fry until golden, 3 to 5 minutes. Use a slotted spoon or frying spider to transfer to a paper-towel-lined plate. Repeat with the almonds, cooking until golden, 3 to 4 minutes. Transfer to the plate and then turn off the heat.
Add the bread to the oil and cook until golden on both sides, about 2 minutes total. Remove and transfer to the paper-towel-lined plate. Fry the garlic until golden, about 3 minutes, then transfer it to the paper-towel-lined plate.
Transfer the hazelnuts, almonds, bread and garlic to a blender. Add the water, piquillo peppers, tomato powder, vinegar and salt and blend on high speed, adding the olive oil in a slow and steady stream to emulsify and create a thick mixture. Pour into a medium bowl and serve with grilled vegetables and goat cheese or refrigerate in an airtight container for up to 5 days.