Rough chop the strawberries & add to a pot over medium heat with lemon juice, zest and sugar. Cover and let simmer, gently mash down some of the strawberries and let cool.
In a food processor or blender, add the chopped beets + sugar + water, pulse until smooth.
In a large bowl, add the water, dry yeast & sugar. Stir until sugar is dissolved & let it stand for about 5 mins or until it gets bubbly. Add the flour, salt & strawberry + beet mix to the bowl & stir until well combined. If too sticky, add more flour 1 tbsp at a time until a dough ball forms.
Knead dough for several minutes & leave in a oiled bowl covered with a damp towel. Let rise until it doubles in size, about 1 hr. Punch the dough and cut into 8-10 equal pieces, roll into a ball and cover again with the towel to let rise for 30 mins.
Poke a hole in the center of each risen dough ball to form the bagel shape. place a parchment paper square under each bagel & allow these to rise once more for another 30 mins under the damp towel. While these rise, prepare your poaching water, bringing water + brown sugar to a gentle boil.
Using a spider or a slotted spoon, place a bagel including the parchment paper into the simmering water (2 will fit in the pot at once), poach each side for 1 min and set on a baking rack (keeping the parchment square on the bottom) once removed from the pot while you poach the rest.
Using a pastry brush, mix the egg + 1 tbsp water and coat the bagels in the egg wash.
Arrange bagels on their parchment squares on a baking sheet & bake at 415 F for 20-25 mins; until very lightly browned.
