Whisk together the olive oil, lemon juice, sherry vinegar, mustard, salt, and pepper until emulsified. (Alternatively you can add the ingredients to a clean jar wih a lid and shake until the vinaigrette is well blended.)
Shred the trimmed Brussels sprouts using the shredder attachment on your food processor or by hand on a box grater.
Toss the sprouts in a bowl with the lemon vinaigrette, then divide among two serving bowls.
Sprinkle the pecans and shave parmesan cheese over each of the Brussels sprouts bowls.
Top each bowl with two poached eggs each and grate with fresh pepper.