Finely dice the strawberries (should be about 2 cups) and mix them with the lemon zest, lemon juice, and brown sugar. Stir occasionally, pressing on the solids with the back of a spoon or a potato masher to draw out the juices.
In the meantime, add water to the bowl of a standing electric mixer, sprinkle the yeast and a pinch of brown sugar over its surface, and stir to combine. Let sit for 10 minutes until it begins to bubble.
Add the macerated berries, one cup of flour, and the salt to the yeast mixture. Mix with the paddle attachment or a wooden spoon until well-combined. Add the remaining flour, about 1/2 cup at a time, to form a thick dough that clears the sides of the bowl. Change to the dough hook and knead for about 5 minutes. The dough should feel slightly tacky but not sticky.
Place the dough in a lightly greased bowl. Flip the dough to coat completely with oil and cover the bowl with plastic wrap. Let rise until about doubled in bulk, about 1 to 1 1/2 hours.
Transfer the risen dough to a lightly floured work surface and divide it into 12 equal pieces. Using just enough flour to prevent sticking, shape the dough into tight, round balls, pinching and smoothing the seams on what will be the bottom side of each bagel. Cover the balls with plastic wrap and let them rest for 20 to 30 minutes.