1Place the shallots, both vinegars, the measured salt, and the pepper in a medium, nonreactive bowl. While whisking constantly, slowly add the oil in a thin stream until fully incorporated, about 1 minute.
2Place the fennel in a large bowl, season with salt, and stir to coat. Add the arugula, strawberries, and pine nuts and gently toss to combine. Add the vinaigrette and toss to coat. Season with additional salt and pepper as desired. Add the Pecorino Romano, tossing lightly (so as not to break up the cheese) until just incorporated.