Whisk flour, instant yeast, sugar, and salt in a bowl. Stir in cold water and 3 tbsp milk and the 2 tbsp softened butter until a shaggy dough forms. Knead briefly (2–3 minutes) until smooth. Shape into a disk, wrap, and chill 1 hour.

Prepare butter block

While dough chills, place 8 oz cold butter between two sheets of parchment. Pound and roll into a 6x8 inch rectangle about 1/2 inch thick. Keep chilled.

Laminate dough

On a lightly floured surface, roll chilled dough into a 10x8 inch rectangle. Place butter block centered on dough, fold dough over like an envelope to encase butter. Roll gently to 12x8 inch rectangle and perform a book fold (fold one short end to center, other short end over it). Chill 30 minutes. Repeat rolling to 12x8 and single fold two more times with 30-minute chills between (total 3 turns).

Shape croissants

After final chill, roll dough to about 10x12 inch rectangle, 1/8–1/4 inch thick. Trim edges and cut into 4–6 triangles depending on preferred size. Roll each triangle from base to tip to form croissants. Place on baking sheets lined with parchment, point tucked underneath. Chill 30–45 minutes until slightly puffy.

Bake

Preheat oven to 400°F (200°C). Whisk egg with 1 tsp water for egg wash and brush croissants. Bake 15–18 minutes until deep golden and layered. Cool 10 minutes before slicing for sandwiches.

Bacon

Preheat oven to 400°F (200°C) if not already. Arrange bacon in a single layer on a rimmed baking sheet lined with foil. Bake 12–18 minutes until crisp to desired doneness. Transfer to paper towels to drain and keep warm. (Alternatively fry in a skillet over medium heat, 4–6 minutes per side.)

Arugula Aioli

In a small food processor or blender, combine 1/2 cup mayonnaise, 1/2 cup packed arugula, 1 tsp lemon juice, 1/2 tsp minced garlic, 1 tsp olive oil, 1/4 tsp salt, and 1/8 tsp black pepper. Blend until smooth and bright green. Taste and adjust salt/lemon. Refrigerate until assembly.

Poached Eggs

Bring a medium saucepan of water (about 3–4 cups) to a gentle simmer. Stir in 1 tbsp white vinegar and 1/2 tsp salt. Crack each egg into a small bowl. Create a gentle whirlpool in simmering water and slip one egg at a time into the center; poach 3–4 minutes for runny yolk or 5 minutes for medium. Remove with a slotted spoon, drain on paper towel, keep warm. Repeat for remaining eggs (work in batches if needed).

Optional: If making multiple sandwiches, refresh poached eggs briefly in warm water (30 seconds) before serving to rewarm.

Assembly

Slice each croissant horizontally. If desired, lightly butter and toast cut sides in a skillet or under the broiler until golden.

Spread 1–2 tbsp arugula aioli on both cut sides of the croissant.

Layer 2 slices of crispy bacon on the bottom half of each croissant.

Place slices of half a ripe avocado (season lightly with a pinch of salt and pepper) over the bacon.

Top avocado with a small handful (about 1/4 cup) of fresh arugula.

Carefully place a warm poached egg on top of the arugula. If you want the yolk to act as sauce, gently press the yolk to test doneness.

Spread additional aioli on the top croissant half if desired, then close the sandwich and serve immediately. Serve with extra aioli on the side.